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.
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.
* 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.
Look on the strength training as grinding out some tasks in an MMORPG to level up an ability. +1 strength every 2-3 months.
Also, as you get older, having denser bones from lifting while younger helps offset osteoporosis.