How do I understand White's +6 advantage here? [White "White (to move)"]
[Black "Black"]
[FEN "rnq1kb1r/ppp3pp/4pn2/4N3/2QP4/2N2P2/PPP3PP/R1B1K2R w KQ - 0 13"]


In this position, White can still castle, but Black cannot (having moved the king twice). Stockfish 10+ (depth 18) assures me that White has an advantage of +6.1, but I am unable to really see why the advantage is so great. I tried some continuing lines of play but Black can hold on to roughly equal material for a long time, so I do not even know how much of the advantage is due to a future tactical threat or merely a superior position.

I am hoping that there is some simple reasoning that is humanly understandable and does not require looking at all possible lines of play for the next 10 moves. Anyone can help?
It really is about king position, but also about how it affects the evaluation when it programmatically adds up the and compares the differences in mobility for each side's pieces. This is where you get the huge difference, and I think this is what you are looking for.

In each of the following lines, deemed best by Stockfish 11, you can see how little the black pieces can move. in particular, it is of note that in each case, the Ra8 and the Qc8 hardly move, and when the Qc8 finally does, it is just to sacrifice itself, or loses material on the move. The same for the Ra8 in the main line that follows. By comparison, look at the mobility of every white piece. The difference is enormous.

For example:

 [FEN "rnq1kb1r/ppp3pp/4pn2/4N3/2QP4/2N2P2/PPP3PP/R1B1K2R w KQ - 0 1"]

 1. Bg5 Be7 2. f4 Nbd7 (2... c6 3. O-O Nbd7 4. Rae1 Rf8 5. Nd3 Nb6 6. Qb3 Nbd5 7. f5 exf5 8. Nxd5 cxd5 9. Nf4 Rf7 10. Bxf6 gxf6 11. Rf3 Kf8 12. Rg3 Bd6 13. Ne6+ Qxe6 14. Rxe6 Bxg3 15. Qa3+ Kg8 16. Qxg3+ Rg7 17. Qd6) (2... Rf8 3. O-O Nbd7 4. Rae1 Nb6 5. Qd3 c6 6. f5 exf5 7. Rxf5 Kd8 8. Rff1 Qe6 9. Ng6 Qc4 10. Nxf8 Bxf8 (10... Qxd3 11. Ne6+ Kd7 12. cxd3) 11. Rxf6 gxf6 (11... Qxd3 12. Rxf8+ Kc7 13. Bf4+ Kd7 14. Rf7+ Kd8 15. Bg5+ Kc8 16. Re8#) 12. Bxf6+ Kc7 13. Be5+ Kd8 14. Qxh7 Nd7 15. Rf1 Qb4 16. Qh4+ Kc8 17. Qg4 Kd8 18. Qh4+ Be7 19. Qh8+ Nf8 20. a3 Qxb2 21. Rxf8+ Bxf8 22. Qxf8+ Kd7 23. Qf7+ Kd8 24. Bf6+ Kc8 25. Qe8+ Kc7 26. Qe7+ Kb6 27. Na4+) (2... Nc6 3. O-O Nxe5 4. fxe5 Nd5 5. Nxd5 exd5 6. Qxd5 Bxg5 7. Qf7+ Kd8 8. Qxg7 Be3+ 9. Kh1 Re8 10. Qf6+ Re7 11. d5 Bd4 12. Qg5 Ke8 13. Qh5+ Kd7 14. Qg4+ Ke8 15. Qxd4) 3. O-O Nb6 4. Qd3 Nfd7 5. Bxe7 Nxe5 6. dxe5 Kxe7 7. f5 Qd7 8. Qg3 Ke8 9. Rad1 Qf7 10. fxe6 Qe7 11. Ne4 Rd8 12. Nd6+ cxd6 13. exd6