What does “Press the point” exactly mean?

Press the point in this case means:

Keep making an issue of this point.

or

Continue bringing this point to people's attention.

The point they are referring to is the hiring of non-white models. It is not a physical location, but rather a "talking point" or an issue that is being discussed.

It's point as in the expression, "Get to the point."

Webster's New World Dictionary, press, transitive verb

9. to lay stress on; be insistent about; emphasize

It may also have come from sports. In tennis, a player may rush to the net to increase pressure on the opponent, trying to win the point. It means increasing the pressure in your efforts to win. So it implies a battle is being fought against opposing forces. Consider:

"Don't press your luck."
"Don't push it!"

It can also carry a connotation of lack of regard for politeness, since pushing "uncomfortable" issues when others are trying to avoid them is considered rude.