In the original Dungeons & Dragons Adventure Game (1974), there were four races - Human, Elf, Dwarf, and Hobbit. There were also Ents in the monster listing.
When the Tolkien estate sued for copyright infringement, the game's publisher changed the names from "Hobbits" and "Ents" to "Halflings" and "Treants" respectively. Since your question was about halflings, I'll stick to those. I would imagine the thought process went something like this.
1. We want a race like Hobbits in D&D.
2. We can't use the name "Hobbit" because the Tolkien estate won't let us.
3. Hobbits are referred to as "halflings" in "The Lord of the Rings."
4. Halfling isn't copyrighted.
5. Ergo, we'll change the name of our hobbit race to "halfling."
As originally conceived, D&D halflings resembled their LotR forebears totally, down to the hairy bare feet. As the years went by, variant halflings kept cropping up in various settings, and come 3e, D&D went from more hobbit-like halflings to the sort of nomadic kender/warrow mix we have now. Probably it was done in part to further differentiate them from gnomes.
Hope that helps.