Consider the following examples
1. OR: You are in corssroad and you can follow 3 directions (left, right or straight). You can choose each with probability $1/3$. What is the probability that you will choose left **or** straight? Obviously $1/3+1/3=2/3$.
2. And: A school has $40\%$ boys and $50\%$ of boys like soccer. What is the probability that a randomly chosen kid is a boy **and** likes soccer? Obviously the boys that like soccer are the $20\%$ of the total population of kids in the school, since they are the $50%$ of the $40%$ which is equal to $50\%\cdot40\%=20\%$. This is the required probability.
The intuition behind this examples is that OR means more choises so you add up probabilities, thus $+$, but AND means more restrictive characterization (you have to be that and that...), thus $\cdot$. Note, to this, that multiplying two nonnegative numbers that are less than $1$ you receive an even "smaller" number (still nonnegative and less than the other two).