Board Game Arena

There is an eye button next to the stratagem. When clicked it flips the stratagem face down. When clicked again it returns to being face up.

It is highlighted when the stratagem is face up, not highlighted when the stratagem is face down.

What is it actually doing? Does this interact with the "Ask to Play Stratagem" Option and this controls if I'm getting asked to play or not?

It works the same way as facedown legions, it allows you to see the hidden face of the card. But as it's useful information, it's activated (outlined in orange) by default, when Legions eye needs to be activated manually.