At its core, GPS works by having each satellite continually broadcast the current time. By observing the time-of-flight differences from multiple satellites (at least 4), the receiver can then compute its position and the current time.
There's a lot more to it, of course, in terms of how it's encoded and how you know where the satellites are and such, but that's the basic principle.
Note that the receiver itself doesn't know what time it is until it gets a position lock from the satellites. Receivers can have onboard clocks, which helps them calculate their position and the time more quickly than starting up with no idea of the time at all, but it's just a helper, not a requirement. Even if you have one, the onboard clock will necessarily be much less accurate than the ones on the GPS satellites, so any (reasonable) delta will be assumed to be clock drift, not spoofing.
So, if I set up a receiver at point A, then rebroadcast the signals I receive to your receiver at point B, and your receiver is blocked from getting the true signals at B, your receiver will think it's at point A. It will also think that the current time is behind, by whatever the rebroadcasting delay is, but it won't have any way of knowing that's wrong.
As far as mitigation goes, if your receiver was running before the spoof signal started to arrive then you could easily notice that things had suddenly gone crazy. I think military equipment that depends on GPS also has backup inertial navigation systems which would quickly show any divergence, and allow it to continue without GPS input (albeit at reduced accuracy).