Build a culture building (Broadcast Center, Media Hub, Temple of Communion) either at or near the planet. Those buildings generate culture, which will raise the allegiance at the planet you put it at, then spread to nearby planets (phase lanes change to match your empires color to show the spread).

That will raise the allegiance to the planets cap, which is shown if you mouse over the allegiance thing with the planet selected. The cap is 110% for your capital, and goes down as planets get farther away. There's only two ways to increase the cap:

1. Advent have a technology to increase the cap.
2. Move your capital planet (its good to have it in a spot where its central to your planets).

If anyone knows the answer to my last question, I'm really curious. Thanks.

In terms of allegiance, the only difference is that it goes up faster the closer your planet is to a source of culture. The cap doesn't change, and any planet within 2 hops of a culture building will get to the cap eventually (so no need to spam broadcast centers).