why move Nf3 to Ne5?

For starters, I'm a noob and realize that I make a lot of mistakes and blunders, but I usually can figure out what I did wrong and act like I'm not going to make the mistake again. I recently finished a 10min game vs licobob where I won, but I'm really confused as to why the engine wanted me to, several times, move Nf3 to Ne5 leaving my queen vulnerable to the bishop on h5. It also wanted me to take Nxe7 from Nd5. Those moves seem like total blunders with not enough strength to win and many pieces lost in the process. I'm sure I'm missing something a child could immediatly see, lol. Idk how to post pictures, but I think you can click on my name and see recent games played too. thanks in advance for any help you guys can provide [Event "TjReece vs. licobob"] [Site "Chess.com"] [Date "2020-11-23"] [White "TjReece"] [Black "licobob"] [Result "1-0"] [WhiteElo "751"] [BlackElo "770"] [TimeControl "standard"] [Termination "TjReece won by resignation"] 1. e4 e5 2. Nf3 Nc6 3. Nc3 d6 4. d4 Bg4 5. Bc4 exd4 6. Nd5 Nf6 7. h3 Bh5 8. Bg5 Ne5 9. Bb5+ c6 10. Ba4 Be7 11. Qxd4 O-O 12. O-O-O cxd5 13. exd5 Bxf3 14. gxf3 Nxf3 15. Qe3 Nxd5 16. Qxf3 Bxg5+ 17. Kb1 Nb4 18. h4 Bxh4 19. Rxh4 Qxh4 20. Kc1 Rac8 21. Qxb7 Qf4+ 22. Kb1 Rfe8 23. a3 Kf8 24. axb4 Qxf2 25. Rxd6 Qf4 26. Bxe8 Qf1+ 27. Ka2 Rxe8 28. Qc7 Re1 29. Rd8+ 1-0

 

You are able to win the black queen and emerge up material after this reasonable complicated sequence (or similar, after Bb5+ Black is losing their queen one way or another). 

ok it's really cool you're able to post like that.

but I end up with a knight, 2pawns, and a queen. black ends up with a queen and knight 12-14. but Then after bxc6+ it appears I'll lose the bishop which ads 3 or 3.2 to blacks total placing them in the lead. I'm assuming the prevention of castling makes it worthwhile?

Continuing for example...

 

Thank you. It helps a lot and you definitely gave me a eureka type moment. I'm a ways off from visualizing that in game, but on the road to 1000 I go! lol.

No worries. This is not an easy tactic to see because you don't win the black queen immediately after giving up your own so don't be concerned that you didn't quite understand why Ne5 was suggested. Who just leaves their queen hanging right? It does show that even straight out of the opening there can be tactics lurking beneath the surface. Got to be on the lookout for them at all times.