The word 까지 implies there is an endpoint to the destination or a limit to the amount of walking being done, so "up to the school" is the most accurate translation. "The Bear Walks To School" would be "곰이 학교에 걸어가요".

There are 3 forms of this word that I know of:

걷다 which is "to walk"

걸어가다 which is "to go by walking"

걸어오다 which is "to come by walking"

Here, the Korean and the English translation are specifying that, rather than the bear simply walking in the direction of the school, the bear's destination is the school itself, and is walking to go there.

So, a more apt situation to just use the simple 걷다 would be:

곰이 길에 걸어요

The bear walks in the street.

Hope that helps :)

Literal translations:

걸어요 - walks

걸어가요 - goes by foot