box
凾 centers on a container: a box or something that holds items. The verb sense of putting something in follows naturally from the idea of a box.
凾 is a variant form of 函, which originally depicted an arrow container. The modern form 凾 replaces the inner 氺 with 了 and 口, but the exact historical development of this variant is uncertain.
The outer shape 凵 is like an open container. Inside, 了 and 口 and 又 suggest items being placed into a box: a hand 又 putting a small object 口 into a finished 了 container.
For カン, imagine a can that is a box: can -> カン, and you put things into the can-box.