The "Proof" not proving anything
Although Moore did this for specific reasons, he called his document a proof, but it's really more just a critique of what he considers to be unethical philosophical practices. He believes that philosophers have an obligation to stay honest and rooted in the ground. That's what he asserts, but those things technically can't be "proven" because they're issues of opinion.
The hands
There is an ironic gesture in the work where Moore famously offers his "proof" by saying, "Look, here's one hand. Here's my other hand." This argument seems to mean that reality is real because it feels real, and because our bodies seem to be real objects.
Perception
Human perception is another ironic problem because it takes itself for granted. A person naturally trusts their intuitions and perceptions, but Moore is saying that it is that way on purpose, and that to ignore the truth of human perception was wrong.
Ontological irony
What exists? There is a deep irony in human experience, because it begs the question of its own reality. It looks real, and it feels real. British philosophers were interested in more metaphysical abstractions about what reality could hypothetically be, and instead of that Moore basically says if it looks real, and it feels real, it's real for all intents and purposes.
Logic and Reason
By analyzing the thought patterns of idealist philosophers in Britain, Moore challenges the blind assumption that any idea which can be discussed through arguments must necessarily be provable. That's not the case, he asserts, and he re-submits his opinion that the only way to judge reality is through human experience. This is a mark against logic, which is weird for a philosopher.
