I don’t use probability distributions in everyday life ;)

But it is the right distribution to represent uncertainty about the probability of binary events (eg a website user clicking some button). For example, if I have absolutely no idea the probability then I use the uniform distribution, Beta(1,1), which is the maximum entropy distribution. Then if I observe one user and they happen to click, I have Beta(2,1), and at a glance I known the mean of that (2/3) which is a useful point estimate.

Proportions of things frequently follow beta distributions. I think of it as the normal distribution of the domain 0 to 1.