I think the dislike for proofs, and the belief that they must not be important, comes from two misconceptions:
- That proofs are all two-column proofs.
- We prove things to find out if they are true.
The first is blatantly false. Most proofs are paragraph proofs, and don't have to show every tiny little step. No one will be displeased with you if you commute a few variables here and there without explicitly saying so. Often, proofs are taught in high school with a very specific format, and straying from it is penalized. This makes it seem pretty arbitrary and dumb.
The second is a little more subtle. Proofs do, in fact, prove that things are true. One benefit of that is that we can be absolutely certain we're working "on solid ground", so to speak. But if you aren't sure if something is true, you don't go trying to prove it first. The more important aspect of a proof is that it is a justification as to why something is true. Proving the "why" of something gives you
- the potential for more generalization
- a larger toolbox for proving other things
For the first point, if you notice Möbius inversion works for the totient function, you could just prove that and call it a day. But if you look at the proof, you might notice that it doesn't actually involve computing the totient function at all, and it just uses the fact that it's multiplicative. (Well, more likely it'd take a bit of needling to give up that fact, but it's still possible to pry it out) This gives you a far more general theorem, one that would be much harder to find if you only knew the special cases, without proof.
For the second point, anorton's answer has a very good example involving double and triple angles.
Ultimately, proofs are for verifying results that you're pretty sure are true. Because everything you're taught has been proved, this seems useless for quite some time. But when you actually do need to find new things, proofs are indispensable.
The other comparison I've heard is to science labs. Yeah, why go and test that $F = ma$ is true if we can just check the physics book? Because that's not how science is done out in the real world. You use it to support hypotheses that you have. (Although in math, you get to prove them!)
On the other hand, this can be a dangerous analogy, because this is exactly the opposite of how math works. You don't conjecture things on paper and prove them in the lab, you conjecture with numerical examples (the lab) and then you prove things on paper.