So, at 36 years old I'm flabby around the middle and cripplingly weak with sticks for arms and legs. I also discovered a varicose vein in my right leg, which freaked me out.
I discovered Scooby's Workshop (http://scoobysworkshop.com/) through a recommendation from my uncle, started watching his videos, picked up his beginner workout (http://scoobysworkshop.com/beginning-workout-plan/) and have been plugging away with different workout plans ever since. Took me about 5 months to start with before I could do a single pull-up.
I've now been doing something based on Starting Strength (by Mark Rippetoe - https://www.amazon.com/Starting-Strength-Mark-Rippetoe-ebook...) for about 18 months. (If you're a bit older I don't recommend you follow his dietary advice too strongly in terms of Gallon Of Milk A Day, but you may need to up your food intake to lose weight in a controlled way - do NOT try to lose more than 1kg/2lb per week or you may start to lose muscle mass as well as fat.) It's been a bit of a struggle with the new client I'm working with, so it's a case of exercise when I can. I've had a couple of injuries that have slowed me down as well, but progress is still occurring. I've gone from being as weak as a kitten (struggled to deadlift 25kg) to the point where on a good day I can deadlift 150kg for 5 reps. My aim is to get to 200kg.
Bottom line is the only real way to lose weight and keep it off is to increase your metabolic rate, and that means strength training to increase your muscle mass. The problem with a diet and cardio only approach is that you can end up atrophying your muscles, which drops your metabolism, and thereby makes it easier to pile weight back on again.
(Being strong also has the benefit that it just makes life in general a lot easier, and will increase your confidence.)
HTH.
* Building muscle mass while you're young helps you stay mobile instead of becoming frail when you're old
* It feels nice to be able to lift heavy stuff.
The strength feels good, gives confidence, and keeps you healthy. (Especially back injuries will be less likely. And for women fighting osteoporosis is always an additional concern.)
As for looks: you won't magically get a six pack and bulging muscles. I deadlifted 160kg and squatted 170kg (yeah, my grip sucks..) at a bodyweight of 70kg, and I'm no hulk.
But, the extra muscles will help you carry whatever fat you have better.
Also, as you get older, having denser bones from lifting while younger helps offset osteoporosis.
This can be as prosaic as doing DIY work around the house, working in the garden, going shopping, helping my parents move house, pushing my motorcycle around, etc. It also makes holidays involving sports or physical activities (skiing for example) much more enjoyable - you can do more, and push yourself harder, and you have more confidence in doing it.
The routine I'm doing does not make you look like a bodybuilder - as others have pointed out, it makes you strong. This does involve some increase in muscle mass, but I don't think I look particularly big even though I'm much stronger than I used to be.
The last reason is very personal: it's frank curiosity. I grew up as the weakest kid in the class, one of the weakest in the school. I didn't think I could be strong, so when I realised I was wrong, I wondered how strong I could become. It's obviously a harder journey because I'm older, but still I don't feel like I'm near the limits yet.
Look on the strength training as grinding out some tasks in an MMORPG to level up an ability. +1 strength every 2-3 months.