Here is the image of the question: probability
It is written that we need to take probability of occurrence of at least one of A and B. Shouldn't it mean an occurrence that belongs to both A and B and is occurs at least one?? I am all confused by the statement of the question.
I am actually confused as to what your precise confusion is :p ...particularly this question:
If it helps visualize things...
As far as what the blue text in your image is saying, note that the probability that $A$ or $B$ occurs is one minus the probability that neither occurs:
$$P(A\cup B)=1-P((A\cup B)')=1-P(A'\cap B').$$
If $A,B$ are independent, then $A',B'$ are also independent (why?), and thus
$$P(A'\cap B')=P(A')P(B').$$