This was a really interesting puzzle and was enjoyable to work! $+1$ from me indeed! It seems like moves $3$ and $13$ are heavily interchangeable, so long as they don't prevent checkmate down the road. Again, very nice puzzle!
The moves on white's half of the board are in algebraic expression form for chess. You have to look at it from the perspective of:
Queen to F4, Captures Rook, Check. = QxF4+
This is more in the realm of mental chess where no board is needed; I have always wanted to attempt it against someone, and this is pretty close! So if I could I would give more up-votes!
I believe that black is a novice player and isn't really thinking too far ahead with his/her movements. Thus leading to an eventual defeat dealt by white.
My theory (White :: Black):
1. Knight to F3 :: Pawn to G5
2. Knight to G5 (Captures Pawn) :: Pawn to D5
3. Knight to F7 (Captures Pawn) :: Bishop to G7
4. Knight to D6 (Check) :: King to D7
5. Knight to F5 :: Pawn to D4
6. Knight to D4 (Captures Pawn):: Queen to F8
7. Pawn to C3 :: Bishop to D4 (Captures Knight)
8. Pawn to D4 (Captures Bishop) :: Queen to F2 (Captures Pawn, Check)
9. King to F2 (Captures Queen, Release Check) :: Knight to H6
10. Queen to A4 (Check):: Pawn to B5 (Release Check)
11. Queen to B5 (Captures Pawn, Check) :: King to D8 (Release Check)
12. Queen to D5 (Check):: King to E8 (Release Check)
13. Pawn to D3 :: Rook to G8
14. Bishop to H6 (Captures Knight) :: Rook to G7
15. Bishop to G7 (Captures Rook):: Bishop to D7
16. Queen to D7 (Captures Bishop, Check):: King to F7
17. Queen to F5 (Check):: King to G8
18. Queen to F8 (Checkmate)
Special thanks to @ExcitedRaichu for pointing out that move $3$ was the biggest incorrect move, followed by move $13$.