“S’occuper” and “s’occuper de qqc ou qqn” are two different verbs; so you can’t have “occupies myself” and it must be “who is taking care of”.

The meaning is then “After school, I’m the one taking care of my brothers.” or put in a more usual way “After shool, it is me who takes care of my brothers.”.

PS Always check a good dictionary, such as for instance the TLFi; that could provide those clues to you already and in any case it would start teaching you how to read it and progress in the way of extracting much information from it yourself.

You have to identify je m’occupe de as the reflexive form of a verb.

You seem to have got so far as to identify you are dealing with a reflexive verb but the problem might be that in English the verb you need is not reflexive.

And you might be missing that in order to look it up in a bilingual dictionary the de is necessary. So you have to look up for s’occuper de. And that’s what you find:

s’occuper de
[club] to be in charge of
; [affaire, question à régler] to deal with
; [malade, invité] to look after
; [clients] to serve
; [politique] to be involved in

From those you should be able to deduce which one is relevant for your sentence.

As you progress you will find that French has a lot of reflexive verbs that are not reflexive in English. And you will need to know some of them quite early as you learn. For instance s’appeler (I’m called)/my name is..), se lever (stand up/get up), s’habiller (get dressed), se fâcher (get angry), s’ennuyer (get bored)… Indeed a lot of French reflexive verbs will end up in English as “get+past participle or adjective”, but you cannot hold it as a rule. And of course sometimes the se and be identified as an action upon yourself like se couper (to cut oneself).