It's a matter of practice, but also a matter of feel: When stepping with the front of your foot you have a smaller contact area, so the pressure is higher than using the whole foot. This increases sensitivity to the rock underneath (extremely important with smaller holds), strengthens the muscles you will need for them, and also sort of gives you more friction is some situations.
That being said, this is mostly bullshit. If you find a foot hold good enough to step with your whole foot, do it. It's not a problem. Also, you will use heels quite often when climbing overhangs, so it's just a matter of avoiding using heels when they are not required - since they are slippery and offer you no feel of the rock. "Rules" like this one are built do deal with the very common situation, mostly at gyms, of people that are beginning to climb and as soon as they find a large foothold, place their feet (sometimes even horizontally) as if this were safer or offered you more control. It doesn't. The point is that, after you know how to climb and to judge what is the best possible strategy, just do it. There's no "right" way. I have seen stuff like resting using the head to balance, using knees to mantle, placing gear in grass, etc. It's ugly, but if it works, who cares? You're the judge.
If you're beginner, it's best to practice stepping with the front of the foot, though... ;)
Edit: Since my answer is the most visible one, I will reproduce here two comments by @josh and @dwizum to @imsodin's (very good) answer: the foot is a lever, and standing on your tiptoes gives you maximum reach. This is precisely the reason why my answer makes sense, being the mechanism responsible for maximizing applied pressure: By locking your joints to achieve maximum reach, the tiptoes naturally become the most sensitive and precise part of the foot. Since josh and dwizum didn't post answers, I believe their comments should be up-voted by anyone who reads this answer.