My intuition says that it "weighs" both the current proportion of success "x" and current proportion of failure "(1-x)": f(x;alpha,beta) = constantx^(alpha-1)(1-x)^(beta-1). Where the constant is 1/B(alpha,beta). Alpha is like a "weight" for success's contribution. Beta is like a "weight" for failure's contriubiton. You have a two dimensional parameter space (one for successes contribution and one for failures contribution) which makes it kinda difficult to think about and understand.

My intuition says that it "weighs" both the current proportion of success "$x$" and current proportion of failure "$(1-x)$": $f(x;\alpha,\beta) = \text{constant}\cdot x^{\alpha-1}(1-x)^{\beta-1}$. Where the constant is $1/B(\alpha,\beta)$. The $\alpha$ is like a "weight" for success's contribution. The $\beta$ is like a "weight" for failure's contribution. You have a two dimensional parameter space (one for successes contribution and one for failures contribution) which makes it kind of difficult to think about and understand.