I worked in the art world for years. First as a studio assistant, then as an art prep for large galleries, then eventually I ran my own art fabrication business with a few employees.
A good gallery--emphasis on good--does a lot more then simply sell art to the highest bidder. They strategically place artists into 'important' collections, work with museum curators to bring the artists into a more critical narrative, get the artist into group shows and fairs internationally to contextualize the artist in a current scene or trend, collude with art critiques and magazines, and generally help to promote the artist's career over the long term.
All of this sounds somewhat silly outside the art world, but you have to remember that this is a very particular industry based around historicizing high-brow cultural production. When a good collector buys a painting, they are doing more then just buying a physical object they like. They are throwing their own clout behind the artist and saying "I think what this person is doing is important and I stand behind it."
All of this is done with the intention of increasing the profile of the artist which benefits--each in their own way--gallery, the artist, and the collector over the long term.
As far as the sales split between gallery and artist, standard split is roughly 50% with some variance around material expenses and whatnot.
Note, everything I am saying is the sort of ideal story and there are a lot of bad actors in the business. In reality I find the art world rather gross and the premises it is built upon to be deeply flawed.