Depending on your model of TV and how it's connected, it's possible for the Apple TV to be on and responding to inputs even if the TV is off. If it was on, and a promotional popup came up offering the free trial, then depending on just what the flow is for those popups, it's possible that a small child just fiddling with the remote could accidentally have accepted it.
(I know that our TV is often off while our Apple TV is on; I have no knowledge of how those promotional popups work, as I haven't accepted one.)