Well, in Western lands the land was owned by the federal government before they even were states. In other places the land was paid for - admittedly inadequately.
And for those concerned about the constitutionality of the government owning land, the government owned land before the constitution was even written.
Indeed Amendment 5, contemporaneous with the constitution, anticipates government ownership of land for public use. There is no more widely supported government use of land than the national parks.
Your third grader, who enjoys his playground, would understand the issue far better than your cramped soul.
I know a lot of conservatives and libertarians who are quite content to spend weekends enjoying recreation in our public parks.