Board Thread:General Discussion/@comment-24.6.23.56-20140306041724/@comment-6697950-20140306130831

Even the anime portrays (subtly) that they started developing feelings for each other far earlier in the game. The books of course put more emphasis on such details than the anime and it does explain that Kirito was a source of several huge changes in Asuna's view on life (e.g. from a suicide player who doesn't leave the dungeon to continue battling for her freedom or death to someone who begins to value their life and eventually becomes the head of the clearing group and later from being overly obssesed with clearing the game to realising that even life on Aincrad could be meaningful).

Logically, someone who made your life seem much better would be dear to you. And losing such a person would make you feel despaired.