Simply, it is strength that allows you to do things.
But let's step back for a moment and talk about what it isn't. It is not "body building". Body building is focussed on form. Each body part is sculpted to it's aesthetic ideal. So what's wrong with this approach? Well nothing, if you want to shave you body, cover yourself in oil, and pose for other people. If you want to be functional, then we have problems.
The first problem is training muscles in isolation. Your body is a system. If you want to be functional you need to train it as a system. Functional training concentrates on movements, not muscles. For example, doing a movement like the deadlift works your legs, upper and lower back, and forearms (for gripping the bar). Doing an isolation exercise like a curl works your biceps. A deadlift is functional - picking up heavy shit is something the body was designed to do. A curl isolates the biceps in an unnatural way and develops it out of context with the rest of the musculoskeletal system. Isolation training is not functional.
The second problem is one of attention. People like the muscles they can see in a mirror - the pecs, the biceps, the abs, and the quads. The irony is that these muscles are not as functional as the one's you can't see in the mirror - the upper and lower back, the glutes, and the hamstrings. These are the muscles that help you the most in doing things that require strength. These unseen functional muscles are often referred to as the "posterior chain". "Posterior" being on the backside, and "chain" being a system of muscles that all work together.
Poor posterior chain development is one of the plagues of modern man. Sitting in chairs for 8 hours a day weakens these muscles and leads to knee and back problems. Sound familiar? Of course we all have to work our white collar jobs to pay the bills so everyone needs to train this area of their body. It wouldn't get any work otherwise. If you were to go to the gym and do some bench presses, leg extensions, and some curls, you basically missed the posterior chain entirely. You neglected the part of your body that needs the most attention by fixating on muscles you can see in the mirror when you're posing.
Functional training means training for function and not form. We don't worry about how we look. We worry about what we can do. Luckily for us, form follows function. If you train for function your form will be pleasing. You'll look healthy and in balance since you train your body in concert with the way it was designed. For example, racing cars are designed to be fast, but that demand for function makes for some pretty cool looking automobiles. The reverse is not true. Function does not follow form. If you set out to design the coolest looking car in the world it will not be fast. It may not even be usable.
So be functional. Train the big movements - the squat, deadlift, and overhead press. Then you'll look strong because you are strong.