The game moves are the ones not in parenthesis (normal pgn notation applies) and all the stuff in parenthesis are just variations. It can get confusing when you get lines within lines, but that is all there is to it. I won't describe the entire game with variations here, but let me do the start so you get the hang of it:
The first 4 moves is easy to read until 4. Ng5 when variation(s) follow. "Skip" all the parenthesis to see the next game move. We can then realize that in the actual game it continued 4. Ng5 Bc5 5. Nxf7 and so forth. The parenthesis part we "skipped" is a single variation so that means the variation is 1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. O-O (not the real game 4. Ng5) Nxe4 5. d4 d5 6. Bb5 exd4 7. Nxd4 Bd7.
Alternatively, you can just copy and paste the entire game (with all its variations) into a chess.com pgn diagram and then use the arrow keys to scroll through the game as well as any listed variations:
I understand the basics of reading and writing chess notation, but when I see YouTubers writing down all the different variations in parentheses in their video descriptions I get very confused. For example, how do I interpret this notation showing variations of the Traxler?
1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. Ng5 (4. O-O Nxe4 5. d4 d5 6. Bb5 exd4 7. Nxd4 Bd7) 4... Bc5 5. Nxf7 (5. O-O) (5. d4 d5 6. exd5 Nxd4) (5. Bxf7+ Ke7 6. Bd5 (6. Bc4) (6. Bb3 Rf8 7. O-O d6 8. Nc3 Qe8 9. Nd5+ Kd8 10. d3 h6 11. Nf3 Bg4)) 5... Bxf2+ 6. Kxf2 (6. Kf1 Qe7 7. Nxh8 (7. Qf3 Rf8) 7... d5 8. exd5 Nd4 9. c3 Bg4 10. Qa4+ Nd7) 6... Nxe4+ 7. Ke3 (7. Kg1 Qh4 8. Qf3 (8. Qf1 Rf8 9. d3 Nd6) (8. g3 Nxg3 9. hxg3 Qxg3+ 10. Kf1 Rf8 11. Qh5 d5 12. Bxd5 Nb4) 8... Nd4 9. Qf1 Rf8) (7. Kf1 Qh4 8. Qf3 Nd4 9. g3 Qh3+ 10. Qg2 Qf5+) (7. Kf3 Qf6+) 7... Qh4 8. Qf3 (8. g3 Nxg3 9. hxg3 Qd4+ 10. Kf3 d5 11. Rh4) (8. d3 Qf2+ 9. Kxe4 d5+ 10. Kxd5 Qd4#) 8... Nf6 9. Nxh8 Nd4 10. Qf1 d5 11. Bb3 Ng4+ 12. Kd3 Nf2+ 13. Kc3 Ne2+ 14. Qxe2 Qd4#