Explanation of probability of forks in the Bitcoin blockchain
I haven't read the paper, and I think the expression is only an approximation, but the basic idea is that this is an inhomogeneous Poisson process.

In order for a fork to occur, some unaware node needs to find a block. This is 1 minus the probability that no unaware node will ever find a block. Which itself means that no unaware node finds a block in 1st second, no unaware node find a block in 2nd second, etc. This probability is the product of the probabilities of the individual seconds (since they are all independent).

Each individual probability is roughly (1-P_b)^(1-f(t)) (if f(t)=0 then all nodes are unaware and the probability is the same as no block being found at all a block at all, if f(t)=1 then all nodes are aware and certainly no block will be found by an unaware node, and in between it interpolates. If it's not clear why the interpolation is via a power, keep in mind that for small P_b, the power is irrelevant since the expression is roughly 1-P_b*(1-f(t)).

The combined probability, as mentioned, is the product of probabilities, which is like (1-P_b) to the power of a sum of (1-f(t)), which is like (1-P_b) to the power of an integral.

If it was up to me I would have written the expression differently, but these are the general ideas.