Like Eddie said, it works because of relative minors.
It changes from major to minor throughout the song.

The mysterious part is obviously the minor (F#)
Then when it goes to A it is an old classical move; go from relative minor to the major (adds mood like triumph, success etc)
Then revert back to the relative minor for the mystery effect again (or sad or however you perceive it in a given piece).

Point is it could be in either the key of F#(m) or A.
My training says instead of where it starts, in this situation look at where it ends.
One instructor told me in this particular minor/major situation to simply hit the right bass note when it ends.
In this case it would be the A .. but since it does technically start in F#m I wouldn't challenge it.
The chords want you to feel the minor/major change (the effect).

What makes this one interesting is the C and G chords, as these don't fit the basic scale and are a more unique change than the F#m to A in my opinion.

However neither is new
1-b3-4-1 is a common progression (A-C-D-A section; play that alone a few times and it becomes familiar) much like A-C-D-E ..

The half step modulation from F# to G isn't new either, but used in an unfamiliar way.
Combining the two creates a mood.
I actually wrote a song using F#m-G-F#m-Em (I added the Em at the end, but same effect basically).

As far as key, I can picture either being correct, but it would be F#m and not F# (implying major).

I like how this guy puts it -

I often hear people discussing music theory as if it were a bunch of rules to be followed or broken. But to me this misses the point.

I like to compare music theory to gravity, and to ask people, “have you ever broken the law of gravity?” It’s a silly question and often gets silly answers, but I hope my point is clear: the law of gravity is not a rule to be followed or broken; it just is.

Like music theory, the law of gravity doesn’t tell you what to do, it merely tells you about cause and effect. Gravity doesn’t care whether or not you drop a rock you are holding; it just tells that *if* you let go of that rock, it will fall. You can choose whether or not to let go based on whether or not you want that rock to fall. Similarly, music theory doesn’t care whether or not you resolve a dominant chord to the tonic or not; if just tells you that *if* you resolve it that way, the result is satisfying in a particular way. You can choose whether or not to resolve it based on whether or not you want that particular sense of satisfaction.

Keep this in mind as you study music theory. No “rule” of music theory ever tells you what you must do; it merely helps you understand the effect of various different things you might try.

source -

/just my thoughts on it
Make your sound your own!