In PostGIS why does spatial index still matter for a table with only points?

The description from Boundless is very simplified. It somehow misses the first important step: The index will be able to filter most points/bounding boxes before running any calculation because of their location in the R-Tree.

R-Trees break up data into rectangles, and sub-rectangles, and sub-sub rectangles, etc. The index will look in which rectangle the data lies, and filters any points that are outside this rectangle, then the same with the sub-rectangles inside that rectangle only and so on...

For a better understanding of Indexes/R-Trees yo might start reading the wikipedia article on that topic.