In set theory, the OR probability corresponds to the union of the considered set.
If you consider two not mutually exclusive events, you can identify four regions in your representation:
The area outside the sets (no events happening);
The area of the first event alone;
The region of the second event alone; and
The intersection between the two events.
The OR probability encompasses all the regions that are not "no-events", hence the single events and the intersection between them. Read more