There are two main sub-topics to the free will debates in philosophy, or three, if you count moral responsibility. But I don’t count it, because people who basically agree about when people have genuine choices and when they don’t, often still disagree about moral responsibility. So let’s call that a separate albeit closely related topic. That leaves two main free will topics, corresponding to two main argument types that advocate the conclusion that you don’t have free will.
The first type diminishes “you” to the point where you don’t exist, or don’t have a will. The second admits (at least for the sake of argument) that you exist and have a will, but claims that you are systematically coerced and enslaved by forces alien to you, so you don’t have a free will. I think the second type is far more influential among the general public, so I will focus on that in subsequent posts. (I’m not discounting the possibility of third+ types, but they’re rare beasts if they exist.)
The “non-existent person/will” argument is plausible only to a few people, mainly those who are in the middle of abandoning a dualistic (“soul vs machine”) view of mind and body, but haven’t quite gotten there yet. To reforming dualists, it often seems like the body or brain can’t possibly have consciousness, will, and so on. But really, either the body does have these features, or else non-dualism has an enormous pile of evidence against it. (Hint: the body does have these features.) As human activity is traced to various brain regions, the fearful (ex-, but not utterly ex-)dualist carves off those parts from “me”. That’s a bad move. As Daniel Dennett says, “If you make yourself really small, you can externalize virtually anything.” Don’t make yourself really small.
A less common reason for the Disappearing Self Trick is a bizarre metaphysics recognizing only the properties of microscopic objects (and/or only micro-particles and not composites) as real. As if there could be no cherry pie unless there were cherry pions: fundamental microscopic particles with an inherently cherry property. Or as if there weren’t large objects composed of small parts. But such restrictive “principles” of what’s real are usually selectively applied, indicating that the real concern is mind-body dualism. Often this is accompanied by accusations that people like me, who find mindful states in the human body, are secret dualists: a feat of projection that would make Donald Trump envious.
And that’s all I have to say about the Disappearing Self Trick. If you need more, Eliezer Yudkowsky has several posts (that’s three separate links) that are pretty good. Up next: an intro to the most influential argument against free will, with highlights for closer scrutiny.