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?

Alpha is a decision threshold of how "sure" you need to be that an observed result is incompatible with the null before you conclude that the null is indeed false. For a typical alpha like 0.05, only 5% of the time will you reject the null when it's actually true - you'll get very few false positives, since you require pretty strong evidence to reject the null. For an uncommonly permissive alpha like 0.5, you'll incorrectly reject the null 50% of the time - you'll make many mistakes by rejecting the null since little evidence is required at that threshold.
Alpha = 0.05 is a typical threshold where most people agree that "statistically significant" is generally useful and not a meaningless statistical fluctuation. But each researcher is free to choose their alpha, which in a way represents how much evidence they require to be convinced - in other words, how skeptical they are. A high-alpha individual will spuriously reject many nulls but will rarely miss a true effect, while a low-alpha individual will find fewer true effects overall, but at a higher rate.
Imagine the Bigfoot enthusiast who accepts a grainy photo as sufficient proof of cryptozoology. He has a high alpha threshold, willing to reject the null with little evidence. He will not, however, ever be proven wrong by claiming that cryptids do not exist when they actually do. In contrast, a scientifically rigorous biologist may have a much lower alpha threshold, requiring more evidence to reject the default state of knowledge that there is no Bigfoot. The scientist's skepticism may lead him to disregard evidence of cryptids that truly do exist, but when the scientist claims that one exists, he's more often correct.
There is not one "optimal" alpha that is universally agreed upon for all scenarios. At the one end, a person who uses a very high alpha threshold regularly makes false findings based on little evidence, but at the other end, a person who uses a very low alpha regularly fails to acknowledge real effects because they deem even strong evidence insufficient. What alpha to use depends on your goals and how you perceive the relative costs of false positives versus false negatives.
Alpha represents a decision threshold for whether you deem an observation to be a result of random chance or a true effect. The question could be rephrased as, why am I allowed to decide for myself how much evidence will be convincing? Although there are common standards, it's a personal choice.
