I thought the tooltip was pretty clear? It’s mostly driven by districts, but number star systems contribute as well. It breaks it down where it’s all coming from. Or do you mean how you increase admin cap? That is from civics, a tradition in Expansion, and a few Society research options.
Unless you’re playing a MegaCorp, admin cap should be seen more a soft cap rather than something you must stay under. It’s mostly there to slow the snowballing that tends to happen in these games, but not punish you outright for expansion, from what I can tell.
Edit: I posted this while you were posting yours. Sounds like you were already familiar with everything i said, so ignore this post. :)