Firstly, suppose that A and B are independent events, then the probability that they both occur is just the product of their individual probabilities:
$P(A \cap B) = P(A)P(B)$.
Inthis situation, $P(A|B) = P(A)$ and $P(B|A) = P(B)$. This means that the probability of A occurring is the same whether or not B occurs, or vice versa.
Conditional probability lets us conceptualize dependent events. When there is some relationship between A and B, like A causes B or vice versa, or they have some common cause, then $P(A) \neq P(A|B)$ and $P(B) \neq P(B|A)$.
Suppose we have a jar of 100 marbles. 70 are red and 30 are green. Moreover, 65 have white stripes, and 35 are plain.
Thus: $P(R) = 70/100$, $P(G) = 30/100$, $P(S) = 65/100$ and $P(N) = 35/100$.
Now suppose there is a bias: suppose that most green marbles, 29 in fact, have a white stripe.
If we take all 30 green marbles out of the jar and put them into their own jar, the "green jar", and draw marbles from just the green jar, the probability is 29/30 that we get a striped one. This is obvious: we have 30 marbles, and 29 have stripes. This 29/30 probability is $P(S|G)$: probability of a stripe, when green is taken for granted.
Since 29 of the 65 striped marbles are green, it means that 36 are red. If we put all 70 red marbles into a jar, and choose from just that red jar, the probability of getting a striped marble is 36/70. This is $P(S|R)$.
Now we know that 29 green marbles have a stripe. So the probability of a striped green marble being chosen from the original red-green jar is precisely 29/100. That is $P(G \cap S)$.
If we plug in the numbers:
$$P(G \cap S) = P(S|G)P(G)$$
we see that it works out:
$$29/100 = 29/30 \times 30/100$$
The 30 and 30 cancel, leaving $29/10$.
It works with S and G reversed also. $P(G|S)$ is the probability of a marble being green, if it has a stripe. If we make a "striped jar" with just the striped marbles, it contains 65 marbles. 29 of them are a green. So $P(G|S)$ is $29/65$:
$$P(G \cap S) = P(G|S)P(S)$$
$$29/100 = 29/65 \times 65/100$$
Now the 65's cancel, leaving $29/100$.