As a non-native, I would say 'sich mit etw. befassen' is more like to occupy or busy yourself with something. Pons-Collins also offers: to work on it, to attend to it, to look into it, to spend time on it ... That's the first sense, similar to 'sich beschäftigen.'
There's another sense similar to 'sich annehmen,' which I'm not familiar with, but it seems to be more like looking after something in the sense of taking it on as a responsibility, like for example children.
OT: Do you have a large printed bilingual dictionary? HarperCollins (what I call Pons-Collins, because they used the Pons word list), Oxford-Duden, Langenscheidt-Muret-Sanders -- any of these should have examples of different senses organized by context that are much clearer and easier to understand than an online dictionary like LEO, where all the meanings are jumbled up together. If you're serious about German, at least one fat dictionary of this type is IMHO totally worth the investment.
One very basic grammar thing you can make a general note of: As a rule of thumb, you don't use 'es' after a preposition. You replace it with a compound word starting with da- and ending with the preposition.
The reason (I think) is that after all these dative propositions, 'es' (nominative and accusative) would have to become 'ihm' (dative) anyway; you can't say 'von es' or 'mit es' at all, it does not compute.
But 'von ihm' is (also) the dative of 'er' (he), and 'from/by him' is actually the meaning that people think of first, by default. So if you want the meaning 'from/by it,' you need another way to distinguish it, namely, 'davon.' And when the preposition starts with a vowel, like 'auf,' you stick in an R: 'darauf.'
von + es > davon
mit + es > damit
auf + es > darauf
an + es > daran
über + es > darüber
unter + es > darunter
et many cetera.
If it helps, you can compare this syntax to old-fashioned words in English like 'therefrom' or 'therewith.' (If not, just don't think about English. *g*)
OT: This should be in any grammar book. If you don't have a beginning German textbook, it might also help to get one. (-: