Why is Bxf7+ the best move here?

 Am I missing a possible checkmate or what? To me it seems like giving the piece away to get the king out, but I'm not sure how to use that to my advantage

After Bxf7+ Kxf7 your knights are absolute beasts. If black plays Nd7 blocking his Bc8 and weakening the e6 square this sacrifice is worth considering. eg. Kxf7...Ng5+ Ke8 or Kf8 you have Nd5 hitting the queen which has not many squares and then Ne6. Blacks king will never be safe, you can play a knight to c7 winning the a8 rook, bring in your queen to threaten mate on f7, basically blacks position is ruined.

The general idea behind Bxf7 (I am not sure if the author meant 9. Bxf7+ or 10. Bxf7+) is that the Black king becomes more exposed, while the subsequent check Ng5+ gains a tempo (the Black king must move again). And as Strangemover described and as DeirdreSkye has pointed out in his diagram, the g5 knight has become a super knight. Whether Black can develop his kingside pieces in time to avoid the disaster is another story, but chances are, White is winning. But, unless you are sure that Bxf7 is winning, be careful of it.

These sacrifices are often made on uncastled kings because the f2 and f7 squares are the two squares which are defended solely by the respective kings from the start of the game

Well defined analysis, didn't consider the Knights as a combination. 

Knights have always been annoying due to their unique movement as compared to other pieces. Two knights are even more annoying.