Intuitively:

NP questions take a long time to answer.

P questions take much less time.

The statement "P = NP" means that any NP question can be transformed into a P question. Answering the P question gives you an answer to the NP question.

So, if P = NP, then we can answer questions that we thought were slow, much more quickly than we would have thought possible.

Proving P = NP would (presumably) give you that method for transforming NP problems into P problems. So you would suddenly be able to figure things out easily that currently are hard and take a lot of computing power.

It's often assumed that P = NP would make cryptographic problems much easier to break, but apparently that's not really the case: http://world.std.com/~reinhold/p=np.txt