How do I level up ability expertise? It's in the proficiencies menu, but it doesn't explain how to level it up.

for the time being you can only level those abilities with experience earned in the Lost Grove, which is the current end game.

it does actually cover it, at the end of the proficiency page help/? popup (found by clicking the red ? in the top right corner of the element attunement box), in the "Additional Information For Higher Level Pets" section, if you ever need a refresher on stuff later.
To quote:
"Pets can earn Expertise by Battling Enemies within the Lost Grove, a Level 60+ Zone. Each Pet will have its own Expertise Rating for every Ability in each Element. As a Pet's Expertise Rating increases for an Ability, that Ability will eventually Rank up to the next Rank; increasing its power in Battles. Each Ability can reach the maximum Rank of 5. Additionally, switching a Pet's Element will NOT reset their Expertise Ratings. Those ratings are retained and will be exactly where they were if you decide to switch the Pet back to the original Element."

(the Maws of Madness activity during Fall Festival also allows for rank exp gain)