Why is the denominator in a conditional probability the probability of the conditioning event? Quite a simple or at least short question: Why is $ \frac{P(A \cap B)}{P(B)} $ divided by $ P(B) $ for the conditional probability?
$ P(A | B) = \frac{P(A \cap B)}{P(B)} $
Random image to visualize: 
I actually would like to make use out of the tree display which I can't grasp at all. Why would I divide by $ P(A+) $ when I had to move along this path?
Sorry for the mix-up of $A+$ and $ B $..


In your tree diagram, if you want to know the probability $P(B+|A+)$  Then you completely ignore the $A-$ section, since you know that that does not occur.  So the probability of $B+$ given that $A+$ occurred is the count of the ways $A+B+$ can occur divided by the total number of ways that $A+$ can occur, that is the combination of $A+B+$ and $A+B-$, which together is $A+$, so you would need to divide by the count (or probability) of $A+$ happening.
It might be easier to think of actual events that you can count the ways it happens.
One example:
Roll a fair die (possible outcomes are 1, 2, 3, 4, 5, 6)
A is the event that the roll is greater than 3 (4, 5, 6)
B is the event that an odd number was rolled (1, 3, 5)
So the probability of A and B is $\frac16$ since 5 is the only number that matches,
but if we know that A happened, then $P(B|A) = \frac13$ since we know that the outcome was 4, 5, or 6.  We can get there by counting (1 out of 3 possibilities) or by the math with probabilities $\frac16$ divided by $\frac12$.  If you do not divide by what you are conditioning on, then you are looking at the joint probability, not the conditional probability.  Also think about the Venn diagram.  You need to divide by the area of the circle that you are conditioning on so that the total area of the sum of the conditional probabilities is 1.
