bottom, box, type of oak tree
椢 centers on a container or enclosure made from wood: a box, a bottom, or a type of oak tree used for such wooden items.
椢 combines 木 (tree, wood) with 国 (country, enclosure), suggesting a wooden container or a type of tree associated with such use. The exact historical development is uncertain.
A wooden 木 box 国: picture a sturdy wooden chest with a country's emblem on it, used to store things or as the bottom of a container. The oak tree meaning fits because oak wood is used to make such boxes.
For カイ, imagine opening a wooden box and finding a kite inside: kite -> カイ, and the kite flies out of the box.