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?

The optimal alpha value depends on many factors of the specific research area. It can't be fixed a priori by mathematics alone, and there's no universal consensus on what the "best" value is, nor can there be. Here's a recent paper in PLoS One that tries to grapple with this issue:

Miller J, Ulrich R. The quest for an optimal alpha. PLoS One. 2019 Jan
2;14(1):e0208631. doi: 10.1371/journal.pone.0208631. PMID: 30601826;
PMCID: PMC6314595.

The abstract:

Researchers who analyze data within the framework of null hypothesis
significance testing must choose a critical “alpha” level, α, to use
as a cutoff for deciding whether a given set of data demonstrates the
presence of a particular effect. In most fields, α = 0.05 has
traditionally been used as the standard cutoff. Many researchers have
recently argued for a change to a more stringent evidence cutoff such
as α = 0.01, 0.005, or 0.001, noting that this change would tend to
reduce the rate of false positives, which are of growing concern in
many research areas. Other researchers oppose this proposed change,
however, because it would correspondingly tend to increase the rate of
false negatives. We show how a simple statistical model can be used to
explore the quantitative tradeoff between reducing false positives and
increasing false negatives. In particular, the model shows how the
optimal α level depends on numerous characteristics of the research
area, and it reveals that although α = 0.05 would indeed be
approximately the optimal value in some realistic situations, the
optimal α could actually be substantially larger or smaller in other
situations. The importance of the model lies in making it clear what
characteristics of the research area have to be specified to make a
principled argument for using one α level rather than another, and the
model thereby provides a blueprint for researchers seeking to justify
a particular α level.

Link to the article.
