The answer "so you can pull up on the pedals" makes good sense, but I don't buy it. I don't pull up on the pedals unless I'm climbing a steep, sudden hill. I suspect many, many cyclists don't pedal with full efficiency on the flats. (This is my gut feeling, I have no proof or numbers on this. I remember seeing a study that agreed with this, but I also recall that it was limited in such a way as to cast doubt on the data.)
The reason I use clips and straps is that a lot of energy is expended simply keeping the balls of your feet centered on the pedals. If you don't believe me, try pedaling exclusively with clips and straps until you're used to them, then remove them. You'll find that your feet keep slipping off the pedals. This demonstrates that you were expending effort to keep your feet on-target.
(The only reason I don't use clipless pedals and
shoes -- which are a much more elegant way to achieve the same effect clips and straps are going for -- is that I don't want to have to deal with an extra pair of shoes, and I want comfortable shoes when I get off the bike.)
An experienced cyclist will make constant, small adjustments to keep their feet in the optimal location on the pedals. Clips and straps keep the cyclist's feet in the best places to deliver the most power to the pedals (the balls of the feet), saving the cyclist that effort. (I have a regrettable habit of pedaling on the arches of my feet, which clips and straps correct.) As a bonus, the straps are there when you turn a corner and then see a sudden hill you didn't have a chance to build up momentum to climb.
Edit: I'm at least half retro-grouch touring cyclist, so take this with a grain of salt! I'm a non-racer, saddle-level-with-the-handlebars-for-comfort cyclist.