So the notes of the chords are (bottom note is doubled an 8ve down:

    A   B   C#  D
    C#  D   E   F#
    F#  G   A   B
    ----------------
    F#m G   A   Bm

Or F# minor, G major, A major and B minor.

This sounds good mainly because of the voice-leading. If you think of each part as an individual voice in a choir, (top, middle, bottom), you can see that they all smoothly connect, and none of the voices overlap.

You're correct that the piece seems to be in F# minor ... as the sequence starts on F# minor and goes to B minor - but the chords in between don't have to be strictly notes from the F# minor scale. They're just passing chords on the way to a destination.

You could use these "rules" to try different options:

1. Start on F# C# A, end on B F# D.
2. Smoothly connect the voices from one chord to the next

So any of the following options could also work (and there are many other possibilities). Although some of the harmony is quite distant from the key of F# minor, it's the smooth voice leading that makes it work musically, while also creating some interesting sounds. Try playing these examples and hear what they sound like.

A   B   C   D
C#  D   F   F#
F#  G   A   B
--------------
F#m G   F   Bm


A   C     Db   D
C#  D     F    F#
F#  G     Bb   B
-----------------
F#m Gsus4 Bbm  Bm


A   B    C    D
C#  D    F#   F#
F#  F#   G#   B
-----------------
F#m Bm   G#7  Bm

So the notes of the chords are (bottom note is doubled an 8ve down:

    (This is not a series of guitar chords, please stop converting them SE!)

    A   B   C#  D
    C#  D   E   F#
    F#  G   A   B
    ----------------
    F#m G   A   Bm

Or F# minor, G major, A major and B minor.

This sounds good mainly because of the voice-leading. If you think of each part as an individual voice in a choir, (top, middle, bottom), you can see that they all smoothly connect, and none of the voices overlap.

You're correct that the piece seems to be in F# minor ... as the sequence starts on F# minor and goes to B minor - but the chords in between don't have to be strictly notes from the F# minor scale. They're just passing chords on the way to a destination.

You could use these "rules" to try different options:

1. Start on F# C# A, end on B F# D.
2. Smoothly connect the voices from one chord to the next

So any of the following options could also work (and there are many other possibilities). Although some of the harmony is quite distant from the key of F# minor, it's the smooth voice leading that makes it work musically, while also creating some interesting sounds. Try playing these examples and hear what they sound like.

A   B   C   D
C#  D   F   F#
F#  G   A   B
--------------
F#m G   F   Bm


A   C     Db   D
C#  D     F    F#
F#  G     Bb   B
-----------------
F#m Gsus4 Bbm  Bm


A   B    C    D
C#  D    F#   F#
F#  F#   G#   B
-----------------
F#m Bm   G#7  Bm