"Tycker om" and "gillar" are forms of the verbs "att tycka om" and "att gilla" in the present.

It doesn't mean "fall in love with", it's similar to "like" or just "love", soonely (it depends of context). It's a phrasal verb. Just "tycker" means "think, guess". "Om" means "about". In the meaning "like", "love" you have to use them together. By the way, it's also possible to use "gillar" instead of "tycker om". There is no difference in meaning.

More generally "like". "Om" is a preposition with many possible translations, but its meaning doesn't really matter here since "tycka om" is a phrasal verb, a combination of a verb and a preposition whose meaning cannot be deduced from its parts. Other languages have these too: Even when you know what "to give" and "up" mean in English, you still have to learn the meaning of "to give up" separately, for instance. Or think about "estar para" in Spanish, which isn't simply "estar" + "para" either. This might help you too: https://www.reddit.com/r/Svenska/comments/38cuj1/prepositions_following_swedish_verbs_and/