The federal government leases access. Slant drilling into protected federal lands is a thing apparently… https://www.nps.gov/parkhistory/online_books/thro/adhi4.htm You could research it further if you're interested. Most federal land isn't protected in that way though.

Also, that map isn't high resolution. A lot of areas that appear solid there contain quite a bit of privately owned land that is surrounded by federal lands; even in National Parks/Forests/Monuments/etc. There are more detailed maps around if you care to look for them, but most of the readily available ones are rather dated.