When I first learned about Bayes Law, it was through algebraically manipulating the symbols in the equation until I'd convinced myself that the equation was algebraically correct.
This is totally different from forming an intuition about why Bayes Law should hold. Here's a more visual/spatial representation, which might be more helpful.
First, consider a space that contains all of the events that could possibly occur in the universe. We'll represent that space in two dimensions, and draw a line around it, and call that space 'U':
Now lets suppose another event (say, that I eat a banana today) we call 'B'. The event B is partially overlapping with event A, and so our universe now has four possible outcomes: A occurs alone, B occurs alone, both A and B occur together, or neither A or B occur.
Now lets say we don't actually know how much of the area of U (the universe) that the region AB takes up, but we do know the amount of area A (the red region) that AB takes up. We can think about this as having 'zoomed-in' on the red region:
If we want to recover the unconditional probability, then we just need to 'zoom out' again to look at the full universe U. To do this, we scale the conditional probability down by the ratio of the red area to the total area:
Now bayes law takes this scaling idea one step further, and asks what the probability of AB is given B. We can repeat our last operation in reverse, zooming in on the blue region:
Now we can combine the two steps, beginning with the conditional probability of AB given A, zooming out to the full universe U, and then zooming back in to just area B to find the probability of AB conditioned on B:
And substituting in our probability notation:
Which is what we'd hoped to find, Bayes Law:
To summarize - in operating Bayes Law, we take the probability that an event will occur when we restrict our attention to the conditional event. We then scale that probability to the independent probability of that event occurring in the entire universe by zooming out from our initial restricted region. Then we zoom in again on a different region, to find a different conditional probability.