Answer

For it to rain during the weekend, it can either only rain on Saturday, only rain on Sunday or it can rain on both Saturday or Sunday. We know the chance of raining on Saturday is 20% (note this is not the same as 'Saturday only') and on Sunday it is 30%. If we were to add these, we'd double-count the chance of raining on both Saturday and Sunday once. To convince yourself why, simply draw the Venn diagrams of the different outcomes:

You can think of probabilities of these events as surfaces of the different sections of the diagram. Denoting the chance of rain on Saturday A and Sunday B, we can find the probability of rain over the weekend as: $$P(A\cup B) = P(A) + P(B) - P(A\cap B)$$. Now we already know $P(A)=0.2$ and $P(B)=0.3$, so how can we find $P(A\cap B)$? We can safely assume that rain on Saturday and rain on Sunday are independent events, such that $P(A\cap B)=P(A)\cdot P(B)=0.06$. Therefore, $P(A\cup B)=0.2 + 0.3 - 0.06=0.44$