In an experiment, why can you select your own alpha level for significance testing? From what I have learned, the alpha level tells you that if you repeat the experiment a bunch of times, then (alpha x 100)% of the time you will have a false positive. I'm confused though on if you're the one who gets to pick the alpha level, then how is this always the case? Like if my alpha is 0.05, 5% of the time I will have a false positive (FP), but then if I want to use alpha = 0.5, then 50% of the time I will get a FP. Why is this?

There are some excellent answers, but none discuss risk.
Often, stats are used for practical reasons.  We can talk about the probability that two people in a particular group weigh within a kilogram of each other, or we can talk about the probability that a vaccine has a potentially deadly side effect.  We may use similar tools, but it should be apparent that making a poor decision concerning the latter has ramifications above and beyond messing up on the former. If you tell the FDA that you tested the RSV vaccine for the side effect of Guillan Barre Syndrome at $\alpha=0.05$, the FDA may well turn your application down (as soon as they finish laughing).
Thus, we can choose alpha based upon how certain we need to be.  In fact, when you present a reviewer with the results of your test and the alpha you used, you tell the reviewer what they need to know to make a decision about whether your statistical methods have an appropriate level of conservatism for a given situation.
