I agree; for me at least, other characters are a good way to get me invested in a story. I guess it might be because it’s easier to get invested in other people (even if they’re fictional) than it is to get invested in a plot that you’ll always be looking at as an outsider. (That’s probably why fish-out-of-water plots – like Keeper of the Sun and Moon – work so well, since the MC is just as much an outsider as the reader is.) That’s not to say that I don’t get invested in plots, but these will often take a while to get started, while characters can be there right from the beginning.
Going on to romance specifically, I’ve always seen it as something very important (which is not to say that it should be important for everyone, just that it is for me). It’s not so much a way that I can be more invested in the character I’m romancing (since I romance the characters I’m already invested in), but rather that it helps me to get invested in my MC, who’s often just a blank slate for me to project my own feelings onto. More precicely: if I can project my feelings about romance onto my MC (by having him fall in love with whichever character I’m most invested in), then it provides a connection with him, which helps me get invested in him. In fact, I often find that a compelling romance (or rather, a romance with a compelling character; the romance itself can be pretty basic) in an IF story will make up for a lacklustre plot (and will obviously enhance an already compelling plot).
This is one of the reasons I dislike dead boyfriends so much: it’s not just losing a character I’m invested in, it’s losing a connection to my MC, and I find myself caring far less about them and their plot going on. (This also applies to traitor boyfriends, and even just fake boyfriends like the shepherd from Champion of the Gods.)
EDIT: As for what gets me interested in boyfriends characters, I’m not entirely sure myself, but it generally helps if the game has several guys to choose from. This is one reason why I don’t really enjoy games with preset SOs: if I don’t like them, I’m far less invested in the romance, and thus my MC.
Well, I was into romance long before I met my (real life 
) boyfriend, and I’m still into it now that I’m with him, so I guess it’s probably just down to individual personalities. 
EDIT again:
Because this was mentioned in a previous thread, I thought I should just clarify it here: I’m not really a fan of pure romance stories, either. (I know I’m difficult… 
) Partly because, as @Eric_Moser said, a true romance story will need conflict. Now, I’m not against this on principle – stories usually need conflict – but in “romantic” stories (even non-interactive ones) it often feels as though the conflict is there just so that the story can have conflict, and not as a natural result of the characters’ interactions. This would be magnified in IF, since even if separate conflicts were written for each RO, it would be very difficult to account for exactly how the player (and thus MC) would act, so injecting conflict (for the story) would feel even more unnatural. (And, of course, constraining the ROs and MC just makes it less likely that I’ll like, and thus be invested in, them.)