How do you make climbing shoes not hurt?

First, the real key to making climbing shoes not hurt is to actually buy the correct size, and to not downsize to the point where you are in unbearable pain.

Sizing down is fine, as long as you know what you are doing and not sizing down too much, and that you actually understand the concept of sizing down and how shoes stretch.

The other crucial key, assuming you have bought the correct size, is to just wear them and climb in them. That really is the best way to stretch them.

There are many different methods of stretching them, such as just wearing them in the house, sleeping in them, showering in them, and many others.

A quick search can show you a lot of crazy methods, which some may work, but, even after doing those methods and having the shoe actually stretch, when you put it on and climb in it, it still won't be comfortable since it was not stretched in the same way that you would stretch it while climbing.

When you climb, your shoes bend and twist in all kinds of ways, ways which can't really be mimicked by just walking in them, sleeping in them, or putting them in the freezer, or by any of the other strange methods. Your feet need to be in them and need to be what stretches them.

Just climb in them, they will stretch and fit your feet much better, and they will be stretched in according to the way you actually use them on the wall.

I remember when I bought my second pair of climbing shoes, the ones I bought to replace my beginner shoes I wore for 6 months. I ordered them online and bought a size and a half (at least!) too small.

They were extremely painful to wear, even to just stand in.
My friend, with many years of experience, kept saying it's fine and that I should just wear them and climb in them, and they will loosen up.

I did as he said, but this didn't really work. They did loosen up a little bit even after climbing in them for just a few minutes, they did feel slightly better, but they were still horribly painful to wear.

The reason is, they were just too small! Climbing shoes should not be too small to begin with, even if you are expecting them to stretch and loosen up.
Also, they were synthetic. Synthetic shoes do not stretch as much as leather shoes stretch, so even if they would have stretched eventually by using them and climbing in them, they would not stretch enough to actually make them not hurt, they were at least 1.5 sizes too small to begin with.

And those are the two main points that I have used since then when buying shoes and making them not hurt:

1. Understand what you are buying
If they are leather, they will probably stretch a lot after using them for a while, so if you buy a smaller shoe size, it will probably be uncomfortable in the beginning, or even slightly painful, but as you use them, they should loosen up to be a great fit, just as long as you don't go down in size too much, they shouldn't actually be really painful.

If they are synthetic, keep in mind that they will probably not stretch too much, and if you buy size 10, they will probably stretch at the most to 10.5. I have bought since then 4 pairs of shoes, all synthetic, and none of them have noticeably stretched. They are still as tight as I bought them.
So keep this in mind when buying shoes, since if you buy synthetic shoes that are a few sizes smaller while thinking that they will stretch after you use them and that they will be a good fit, you may be quite disappointed.

2. Wear them!
My friend was actually right, wearing the climbing shoes does stretch them, even for that session, making them much more bearable throughout the session compared to when you are just putting them on.

Of course, if you downsize synthetic shoes too much, like in my accidental case, wearing them probably won't help much. But if the shoes are properly sized to begin with, then you should be fine with just wearing them and climbing with them, they should stretch and loosen up.