I know that あげる is to give. But I find it weird to interpret it as, I have come to love you and I'm giving it to you. But at the same it seems to also make some sense. Is my understanding of the sentence correct?
If I were a Japanese-learner, I probably would have difficulty understanding this sentence, too.
It is a girl's somewhat condescending (but still lighthearted) way of saying "I will be in love with you." It is almost like saying that you are doing the guy a favor. It implies something like "I will love you more for your sake than for my own."
It is the kind of phrase that you hear more often in fiction than in real life.