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?

When you conduct a statistical test, basically it outputs a p-value that you do not choose.
This p-value ($p$) tells you that if the null hypothesis is true and the test assumptions are met, then you'd see a result at least as extreme as the one you're seeing $p \times 100$% of times, if you were to repeat the experiment a lot of times.
The alpha level ($a$) you choose before conducting the test is not the p-value that the test outputs. Your question seems to assume that $p$ and $a$ are the same, but they are two separate things.
Alpha is a threshold that you choose, and use to decide if the p-value returned by the test is significant or not. Ideally, this threshold should make sense and be relevant.
An alpha level that makes sense depends on the context of the study you're conducting. In particular, you should think about the consequences of choosing an alpha level that is too conservative or too liberal (e.g. would some people get hurt if you incorrectly reject the null hypothesis and then take some decision based on that? Would the study cost too much to conduct if you set a too small alpha level? etc.).
Here is a simple example that hopefully should make things clearer if your confusion came from the difference between $p$ and $a$:

*

*You decide to use an alpha level of 0.05 before conducting your experiment and statistical analysis, based on your knowledge of the consequences that this alpha level would have in your specific situation.

*Then, you conduct your experiment, and after that you conduct a statistical test on the data you collected.

*The statistical test returns a p-value of 0.00001: the result is significant according to the threshold you pre-specified, as
$0.00001 < 0.05$. Had the test returned a p-value of 0.3, the result would have not been significant according to the criteria you pre-specified, as $ 0.3 > 0.05$.

