Why do we add probabilities? I know that there is the addition rule of probability, but I want to understand the intuition behind it. Specifically, why does OR signifies addition in probability theory?

Think about it like this
Lets say there is a $6$ sided dice. 
Now, when asked to find $P(3 \mathbf {or} 6)$, consider what this statement actually means. The probability that either $3$ occurs or $6$ occurs. This is equivalent to adding the chances of $3$ and $6$. Therefore $\mathbf{or}$ represents addition. 
