A waistcoat is a vest, a kind of shirt with no sleeves. It just has holes where the sleeves would normally be. Those are the shirt holes.You can hook your thumbs into those holes. That is what the president was doing.

Hello! A "sleeve" is the part of a shirt, jacket, or garment that covers your arms. So the "sleeve-hole" is the part of a sleeve where the garment ends and your hands go through.