How do they other teams sign superstars under the cap?

some go over

Richer owners are more willing to pay the lux tax
small markets typically don't have that. That's why Mikael Prokorov can overpay

In the NBA between the various exceptions and the ability to S&T, and the ability to go over the cap in signing your own players, there is more flexibility in building a competitive roster when a team is above the cap instead of below it.

Under the cap, you have less access to your exceptions, and a team probably has a less than full roster in order to create the room so you've got fewer assets and moving parts to work in the trade market. Being perpetually under the cap, or trying to clear huge room for FA is generally a losing strategy in the NBA, and is usually a lousy way to build a team.