If you're rocking a Men's size 13 boot, you'll almost certainly need Extra Large bindings. XL bindings are designed specifically for US Men's 12 and up, giving your boot the room it needs to sit properly in the heel cup and under the straps.
Trying to cram a size 13 into a Large binding is a recipe for problems. Your boot will overhang the baseplate, which can cause your toes or heels to drag in the snow when you carve — that'll knock you off balance or catch an edge. The straps may also struggle to reach and secure your boot correctly, which is a safety concern.
One important tip: always double-check the specific brand's size chart before buying. Sizing ranges aren't universal across manufacturers, and some brands run tighter or looser than others. Also, if your boots have particularly bulky soles or an inner/outer volume that's on the bigger side, you'll definitely want to stick with XL rather than trying to size down.