$\beta$ (also called sensitivity or power) is a function of the sample size. It totally makes sense to me when we're performing a regression for example. The intuition is not that clear when we perform a binomial test. Consider the following example taken from this wikipedia page.
Suppose we have a board game that depends on the roll of a dice and attaches special importance to rolling a 6. In a particular game, the dice is rolled 235 times, and 6 comes up 51 times. If the die is fair, we would expect 6 to come up 235/6 = 39.17 times. Is the proportion of 6s significantly higher than would be expected by chance, on the null hypothesis of a fair die?
Let's assume the null hypothesis is wrong. Are we more likely to reject the null hypothesis if the sample size (here 235) is large?