>>282079007
Theory that probably has at least a little validity:
She looks exactly like Satella and there's some magic compulsion he has to love Satella possibly due to some soul fuckery we don't know about. Physically, Emilia is pretty much all his fetishes and top interests in a single person and this is likely to not be random chance. Satella and Emilia also happen to look the same for unknown reasons so the same logic also applies to Satella and Satella is more likely the reason for it.
Now, for in-story reasons:
He came over with a dumb set of expectations, that he was owed a bitch because he's the isekai hero. He was a little more honest about this as a motivation for following her around initially, but its also that he's scared shitless. He didn't want to stop and think about really, truly being alone in this world and somewhat clinged to the first person who didn't just brush him off. Because Subaru is also a stubborn jackass, instead of facing his totally wrongheaded thinking and giving up, he instead copes and comes up with new reasons why he should actually keep trying when his previous reasoning is proven to be full of shit or retarded. Basically he "loves" Emilia because he said he did initially and made a big scene about it and now can't ever go back on that so he keeps coping up new ways about how he loves her when things go wrong. This is somewhat plausible on its own, but it might also be combined with the above theory so that he's more compelled to do it than not but doesn't recognize this.