dnd 5th – How can the Githyanki Commander Supreme move without being substantial?

The Githyanki supreme commander, in his block of statistics, is awarded two types of legendary actions, one of them being the teleport action:

Teleport. The githyanki teleports like magic, as well as all the material at his disposal
wear and carry, to an unoccupied space, he can see within 30 feet
of that. […] (Tome of Mordenkainen's enemiesp. 206)

Simple. However, the description of the action continues:

[…] It also becomes insignificant until the start of his next turn. While
negligible, it can move through other creatures and objects as if
they were difficult terrain. If she ends her turn inside of an object he
suffers 16 (3d10) force damage and is moved to the nearest unoccupied
space.

Since legendary actions can only be taken "at the end of the turn of another creature"(specified in the generic description of Legendary Actions), how can the Githyanki Supreme Commander move (or, in fact, end his turn inside an object) while remaining insignificant if the Inequity ends at beginning in turn?

Remember that he can not teleport into a space that he can not see or in a busy space.