I was wondering if "staram se" means care for as in physically tending to someone(s) or if it can also mean care about, have concern for (emotional, rather than physical). Pedro_42's comment seems to indicate that like in English, the Czech word can have both meanings. Would appreciate it if a native Czech speaker or moderator could comment on this issue.
Nobody replied to this in 2 years, so I'll answer, although you're probably not learning Czech anymore.
"Starám se o zvířata" means I actually tend to animals physically. "I have concern for animals" or "I care about animals" can be expressed in a few other ways, for example "Záleží mi na zvířatech" or "Mám zájem o zvířata", depending on context.
The possible translations I found for "starat se" are "take care; look after; mind; attend; be concerned with"; I think the reflexive sense comes from the meaning "being concerned with" (because it refers back to the subject).
This is not the case but, whenever possible, I think it helps to compare the reflexive verb with its non-reflexive counterpart. For example, "učit se" means "to study", whereas "učit" means "to teach". So, what's reflexive about studying? You're teaching yourself!
Lastly, here's a quote from the book "Czech - An Essential Grammar": "Certain very common verbs are permanently accompanied by 'se', and thus (are) always reflexive, (but) not always with any obvious logical reason, e.g. dívat se ‘to look’, ptát se ‘to ask’ (a question), bát se ‘to fear, to be afraid’".