Why does my baked custard form a skin on it?
You did absolutely nothing wrong; this is completely normal.

When exposed to the air, custards (and other hot milk products, like real hot chocolate made with actual milk) will form a skin as the proteins irreversibly polymerize.   This is analogous to the way a skin will form on the top of a paint can left open.

You can prevent the skin by pressing film wrap or parchment directly onto the surface of the custard, preventing air contact after it is cool enough (tough to do with baked custards, but you can peel the skin if you do not wish to serve it).   Of course, this is a trade off, as you will affect the surface appearance of the custard when you remove the covering.

Side note:  the skin is perfectly edible; some people like it, others do not.