The first step in sympathy and acceptance of pedophilia is the idea that a large proportion of pedophiles do not molest children and never will molest children. A large subset of us pose no a danger.
Even once that point is past, lots of people have reactions like, "Ewwww!", "Yuck!", "How can you be attracted to raping kids?"
Some pedophiles do have a crudely felt and expressed attraction to kids. The equivalent for ordinary guys are the ones who mutter to each other, "Wow, what a nice piece of ass! I'd sure like to do her!" I think such men are rarely rapists, partly for fear of getting caught and punished. But the vast majority also wouldn't rape a woman because however gruff their talk, they know it's wrong and wouldn't hurt a woman that way. Most pedophiles with crudely felt attractions are the same. There are exceptions in both cases, and both are enormously distressing -- sexual abuse of either children or adult women.
My main concern in this post are attractions that are felt more finely. And I'd like to concentrate first on the teenage years, when sexual attraction is felt most strongly and perhaps most purely, and where almost all of us start out having no previous sexual experience.
I believe it is quite common for a girl to reach an age where boys are just amazingly interesting, where they make her heart beat faster and make her tongue-tied, where nonetheless the idea of such a boy sticking his penis into her vagina is gross and disgusting. My intuition is that boys are less likely to feel such a mismatch or to admit it, but I suspect it happens too.
It is rare today in the west for teenagers to be ignorant of the mechanics of sex, but it has certainly happened with greater frequency in other times and places. Girls and boys alike in such situations nonetheless feel those same feelings of fascination with the other sex -- of strong crushes that really are sexual -- with no idea how it would be consummated.
Though I have less knowledge on the point, lesbian and gay teens (who know the basics of heterosexual sex) will also feel this fascination or crush for someone of the same sex, but it will often coexist with a feeling of ickiness at just how same-sex sex is done. Giving oral sex? For boys, giving or receiving anal sex? The attraction to the person can coexist with uneasiness about how sex would actually work.
And finally, sensitive men and women of all ages will often look at or consider an attractive partner, attending to personality, voice, face, how they carry themselves as well as their body. They often feel what qualifies as a sexual attraction. That feeling is often going to exist independently of thinking of the other person's sex organs and how exciting it would be to have sex.
We also have one older ideal of a man's romantic love -- of adoring a woman, loving her so much that the idea of actual sex is not what they would want at all. Nonetheless, this is a manifestation of a sexual attraction.
In all these cases, we can see an attraction to the partner as a whole person. Consideration of the sex act comes later, if it comes at all.
Many pedophiles feel exactly the same way.
For me, a 6-year-old girl can take my breath away. I felt that way about such girls until I was over 50 without ever realizing there was a sexual component to my attraction. But I still was a pedophile -- I had a crush on the girl. Now I realize there is a sexual component, and later, when I am alone, I might have a satisfying sexual fantasy about her. When I am with her, I am aware that any sexual expression is completely out of the question and don't experience the attraction as arousal.
Other pedophiles may feel the sexual attraction more directly, but it is still often a two-step process. Step one: Here's a child, he or she is sexually attractive as a person. That's a pedophilic attraction. Step two: How could this be consummated sexually? It can't be in any ethical way. End of story -- no sexual abuse.
Ordinary men may feel a sexual attraction to a great many women. The vast majority of them are unavailable. A sexual advance would be rejected. The men might fantasize about such women later while masturbating, but are they having fantasies of rape? Not at all. They simple imagine that the woman IS interested and consenting. Such a transformation in fantasy is just as easy for a pedophile.
Hopefully you non-pedophile readers remember what it was like when you first looked at someone of your preferred sex and felt that excitement that was the first sign of your sexual orientation. You might be hard-pressed to explain, justify, or prove just why you find that sex of person attractive as opposed to the other one. Most people are not bisexual. Some like women. Some like men. They can't both be correct in an objective sense as to why that sex is attractive and the other is not. It's just how they're made.
Pedophiles feel that way about children. And hopefully that's something you could consider without as much of a "yuck" reaction.