What is the practice of meditation taught there, and what result in a human life do they have if they practice the meditation?
In terms of benefits, consider meditation a tool to understand your mind and to be better able to understand your drives and motivations, and to learn how to focus your attention on various aspects of yourself. What "result" you get from that depends very much on you and your motivations and desires, but what I feel I get from it is a stronger appreciation of the world around me, better concentration, and better impulse control.
A particular a-ha moment for me was how I feel I started noticing when I started rationalizing to myself why I should buy comfort food. It was a distinct feeling of "looking in" and getting to catch my mind in action concocting excuses for why I should give in and buy that chocolate that doesn't fit with my diet, early enough to make it easier to stop them. In other words it feels like it makes it easier to exercise willpower.
Note there's a lot of opinion and feelings here. There is actual scientific data on vipassana style meditation, but I'm a sceptic, and I know full well that some or all of the benefits I believe I see from it could be placebo, or unrelated to the meditation (perhaps it's just because I get more rest), and I certainly haven't measures them. But frankly, I'm also of the opinion that it doesn't matter all that much - if it's a placebo, it's a damn good one, that makes me feel good.
Try it for yourself and see. It is not a big investment in time.
A good and mostly secular introduction free of "woo" is Mindfulness in Plain English. It is available for free in non-DRM'd digital versions here: http://www.urbandharma.org/udharma4/mpe.html or you can buy the paper version of Amazon.
It is notable for very popular amongst non-buddhists, including skeptics/atheists for it's lack of religious content. There are some scattered references to buddhism, but they are mostly providing context and explanation of the background of vipassana, and to me at least, as a lifelong hardline atheist, were not particularly obnoxious.
The Urban Dharma site also have a lot of other introductory resources - I'd particularly recommend the "Introduction to Meditation" audio series by Gil Fronsdal.
I have not done a 10 day sit as described in TFA, but I've had a passably regular Vipassana-style meditation practice for a few years now, and concentration and impulse control have been two major benefits in my life.
One thing that comes with impulse control, but I find valuable enough to merit mentioning on its own, is that I am now much better at letting go of thoughts. I used to have difficulty with intrusive thoughts--fantasies of violence or humiliation that would play on loop in my mind. Now, they appear, drift once through my mind, and are gone. This also applies to getting songs stuck in my head.
For context, I grew up in the Christian church, identify as atheist, and have only a mild intellectual interest in the spiritual trappings of the tradition from which meditation comes to me.
To me, getting rid of attachments means to reduce the impulses that controls my actions "behind my back". It does not mean not caring, but it means noticing that you care, and noticing how that feels, and noticing what the impulses tell you, and then making a decision based on that information rather than just blindly following the impulses.
Often that means acting differently than the impulses tells you because you recognize that other goals you have are better served by not giving in.
Personally I don't believe that means I am abandoning myself, but on the contrary that I am living up to my belief of who I am and who I should be much better by (slowly) ridding myself of compulsions that made (and makes) me unhappy by getting me to act in ways inconsistent with how I want to be.
A lot of human pain and frustration is down to the inconsistencies of self - how we want to do that coding, but end up procrastinating all day and then afterwards feels guilty about it, for example. Maybe that procrastination was nice, and good for us. But we (or I am at least) are happier when we actively make that choice.
I still don't believe I am good at it, but I hope I keep getting better at it.
You said, "etting rid of attachments means to reduce the impulses that controls my actions "behind my back". It does not mean not caring, but it means noticing that you care, and noticing how that feels, and noticing what the impulses tell you, and then making a decision based on that information"
What role does the fact that you care, feel, and have impulses play in your judgement about whether to perform a given action?
Those facts by themselves do not reveal the matter connected to the cases that you are judging.
That's why I'm trying to confirm your standard of judgement that you apply.
"But we (or I am at least) are happier when we actively make that choice."
What do you indicate by happiness?
You said that it's a tool which can accomplish better understanding. Did you become to understand your mind and what causes you have? You say you have concentration but I'd like to confirm what you can see with your concentration.
In the world, an answer is determined by a question. Like in math, 1 + 1 is a problem that makes the result 2 according to a fixed formula. If someone finds 1 or 3 from the question then it's either their mistake in calculation or the causes (numbers in question) were misreported.
In terms of understanding my mind, you seem to be very focused on absolute, objective outcomes. It is not that simple. What questions would you ask? And how would you measure the outcomes?
A lot of questions you might ask can only be evaluated by answers your own mind gives you, and you often can't tell if those answers are objectively correct or just rationalizations.
What you can do is observe what you feel (physical sensations, or internal feelings), how you react, what you think, what brings out specific thoughts etc. and try to bring more of your actions under conscious scrutiny and control.
To go back to my example about impulse control and chocolate: If my mind tells me I deserve that chocolate, is it right or wrong? There is no objective truth to that, because the only arbiter is myself, and if there are rules that says I shouldn't have chocolate, they are rules I set myself and rules I am allowed to change.
What meditation as a tool helps with in situations like that, is teach you techniques to "catch" your mind in the act of rationalizations to let you slow down and consciously evaluate those thoughts before accepting them. Instead of accepting that "yes, I deserve that chocolate", I might think "hold on, I have decided to diet; I can have that chocolate, but I'll feel disappointed later that I had it". Or I might go "yes, I do actually deserve that chocolate; I've done very well on that diet, and as long as it is just that one chocolate, it is an acceptable trade off". Neither decision is still correct or incorrect.
It might feel trite and underwhelming, but we tend to walk through our lives following hundreds of impulses a day that we don't question, or even are aware of: We go one route instead of another to get somewhere. We pick one outfit instead of another. We pick certain meals. We say certain things instead of others. We smile at someone, or ignore them.
A lot of the time there are explanations for these actions that are plain as day if you pay attention. You can go through your life oblivious to the reasons, and thus not influence them consciously. Or you can pay more attention and more consciously decide which decisions to take.
The latter is not more correct. But in my experience it feels more rewarding, as I am often able to catch impulses for me to act in ways that counteract goals that I have set myself, and change my behaviour.
One thing I noticed, for example, was how I'd impulsively do small things avoid people to reduce social interactions with people I didn't know well. This at the same time as I was trying to improve my social skills. Consciously paying attention helped me catch those impulses as they were happening and override them.
When you understand consciousness, you can see that it operates in a fixed principle. Everything that you've done and everything that happens to you all accumulates in you. Things that are latent within you produce the phenomenon of your mind when you meet a given thing.
In order to see your mind I just check what you can see by asking you about what you say you know.
If someone says they know something, they must also show people the evidence of what they know.
You talked a lot about the effect of meditation on your mind but you don't inform people what result it has in their life. Nor do you fully realize why you are experiencing peace through that process.
What use is concentration? All the facts and phenomena that appear in people are determined by what existed in them from the past. When people get distracted it's because of their karma inside. But concentration is not the way to suppress karma, and nirvana (peace) is only attainable by fully stopping karma. The fact of the matter is that only awakening to what exists can stop karma. You have to be aware of yourself and awake to what it is all the time in order to differentiate between right and wrong. Concentration doesn't affect your ability to distinguish good from bad.
It's a fact that meditation is not how Gautama Buddha obtained Enlightenment. His ascetic practice was different than everyone around him. And after he opened his eyes through his Enlightenment, he never taught people that they should practice concentration, nor not to think. He guided them in the opposite direction and advised that people should not accept something as fact without the process of thorough confirmation. It doesn't mean concentrating on the questions. It means that you check the answers in the living reality.