Independence intuitively means that if you have information for one event that it doesn't effect the probability for another event.
So consider the two events: an even number and a number <= 4. If you know that something is even (2,4) then given that its also <= 4 you don't have any extra advantage of knowing what number it is. If its even then you have an equal chance of picking between 2 and 4 and if its odd again you have an equal chance of picking between 1 and 3.
No matter what the scenario, knowing one event doesn't help in predicting another and this is exactly what independence means intuitively in a probabilistic sense.
If one event did greatly effect the probability then you would get an P(A and B) != P(A)P(B).
The proof for independence is really simple. If B and A are independent then P(A|B) = P(A) and P(B|A) = P(B). Since P(A|B) = P(A and B)/P(B) = P(A) we multiply both sides by P(B) and get P(A|B) = P(A)*P(B) for independent events.