Is there an explanation for what Target State does and the options of Any/Full/Empty? I'm sure I'm missing something and it's really simple.

Also generally - How do I get the teleporter to stop sending when the target teleporter on the target planet has enough items? Is this possible?

The teleporter doesn't "send items" as such. It completely exchanges the source and target inventories.

Target state can restrict the teleportation based on the target's inventory. Full only engages the teleporter if the target's inventory is full, empty only when it is empty, and any does not check.

For example, you could have a teleporter on Vulcanus loading up with tungsten and tell it to teleport to Nauvis when full, but set its target state to "Empty" to ensure that you won't be getting anything back. Or set it to "Full" if your teleporter on Nauvis is also being loaded with resources (e.g., uranium) and you only want to teleport once both sides are fully loaded.