A colleague once asked me: what's the point in doing strength training? He said, and I'm paraphrasing here, that getting bigger and stronger is pointless in modern society, and that since I mostly sit around coding on a computer for an average of 8 hours a day, that it wouldn't benefit me. “You're just being vain”, he told me. “It's useless. You’ll end up with roid rage, smash doors down and become an arrogant douchebag with all those damn muscles.”
According to the Mosby's Medical Dictionary, strength training is:
"a method of improving muscular strength by gradually increasing the ability to resist force through the use of free weights, machines, or the person's own body weight. Strength training sessions are designed to impose increasingly greater resistance, which in turn stimulates development of muscle strength to meet the added demand."
Here are some of the benefits of strength training:
A study of older men and women with arthritis in a strength training program found that their training decreased their pain, increased muscle strength, improved general physical performance as well as other listed benefits. It even compared strength training to be as potent at easing pain as medication. 
*Bone strengthening *
As you get older your bone density decreases and the risk of injury increases. This condition is called osteoporosis.
Strength training is a means to combat this product of aging, especially for post-menopausal women who can lose 1-2% of their bone mass annually.
Due to the increase of bone density, the risk of fractures is also reduced.
Increased metabolism and weight maintenance
Strength training is known to help increase your resting metabolic rate. 
it's been found that individuals who have more muscle mass have a higher metabolic rate, as muscle, being an active tissue, consumes calories while stored fat consumes very little energy. It's been found that strength training can yield a 15% increase in metabolic rate, which as we know is useful for weight loss.
Good for diabetics
Studies have found that strength training has found to help those with type 2 diabetes.
It's shown that strength training boosts how well your body uses insulin, improves the way it uses blood sugar as well as better glucose control.
Those who exercise have been found to exhibit greater quality of sleep. Fall asleep faster, sleep deeper, longer and wake up less often.
Sleep benefits yielded through strength training have also been comparable to utilizing sleep medication, without side effects or expense.
Healthier heart tissue
Strength training has been found to improve cardiovascular health.
Those who lift have a reduced risk of heart disease and an increased aerobic capacity.
Exercise is known to release dopamine, serotonin and norepinephrine. Strength training does just that. It make you feel good. 
From another perspective, hitting a new PR (personal record) and feeling the sense of accomplishment when you can, for example, officially squat 1.5 / 2 times your bodyweight is pretty sweet. As the weeks and months go by, you can track your progress and look back at what you used to lift. Maybe you started out only lifting the bar, but now you're at 100kg+. It helps you realise that long term, small and steady incremental changes can yield a massive difference over a lengthy period of time. That outlook also applies to various aspects of life, not just weight training.
I do believe any exercise will help build mental toughness. That marathon endurance runner? It takes a strong mind to keep going for hours and hours. It's like this: you've got a heavy weight loaded onto your upper back for a squat. You go down into the hole and hit that sticking point. You push back up and gravity, fatigue and that egg-salad you knew was just a tad off, are all against you. You need the strong will and discipline to force that weight back up and onto the rack where it belongs.
You'll look better
Yes, it's true. Lifting weights will probably make you more attractive...as long as you don't build yourself with comical proportions: http://bit.ly/19XixFR
BONUS ROUND - MYTHS OF LIFTING:
Squatting is bad for your knees
It's true, squatting is bad for your knees - if you do them poorly. Squats are actually very good for your knees if you execute them correctly. If you want to read more, check this thread out: http://bit.ly/1C9zO5F or this video to get the basic form down: https://www.youtube.com/watch?v=Dy28eq2PjcM and reasons to squat here too: http://stronglifts.com/squat/#Why_Squat
"I don't want to get too bulky" No, lifting heavy weights won't make you ‘Accidentally-Arnold’ overnight. This thinking seems to be more prevalent in women than men though. Think of it this way: Men have much more testosterone than women, yet even after a lengthy period they're still not as big as some women believe they'll be if they lift anything heavy.
Those women you see on the cover of muscle magazines with the massive delts, blocky abs and masculine face are usually on their nth cycle of steroids. Unless you're thinking of going that route, you won't be able to reach that physique naturally.
Hell, here's a 17 year old 46kg girl deadlifting 128kg and does she look like a monster to you?: https://instagram.com/p/ziM5liy0MN/
"B-b-but I just want to get tooooned"
"Toning" is simply the result from building muscle and reducing body fat. Lifting heavy weights is a good way to reach that goal.
Deadlifting is bad for your back
It is - if you do it poorly. Like I said with the squat-knee myth above, if you do any act poorly it can be dangerous.
"How many steroids does whey have?"
Whey protein is a by-product of manufacturing dairy products. It's safe and does not have any 'steroids' in it. Read more here http://en.wikipedia.org/wiki/Whey and here http://examine.com/supplements/Whey+Protein/
"Ok, but that creatine you're on must have at least four and a half steroids"
No. Creatine is also actually found in foods like eggs, meat and fish. Read about creatine here: http://examine.com/supplements/Creatine/
"*Lifting heavy weights isn't natural"*
This is related to the appeal to nature argument: http://en.wikipedia.org/wiki/Appealtonature
If cavemen/our ancestors/primitive man didn't lift weights then we shouldn't. But why does it matter if primitive man did it or not? People drive, surf, cycle, skydive and such and those aren't natural things that they did either.
That medication you're on? The computers you're using? The chocolate you're eating? Those glasses you're wearing? Are those natural? Did cavemen use those?
Appealing to nature and primitive man is a ridiculous logical fallacy to pull out.
To quote Lord Rippetoe: 'Strong people are harder to kill than weak people and more useful in general.'