"весь" is a curious little word that can act as a pronoun or adjective. I think here it is an adjective and is being used the same way we English speakers use it when we say "we all." I don't know why DL separated those two words because the literal translation works fine as, "We all are afraid."
@MichalaK_EU: Yes, it is the dative case versions of "мы" and "все". An example in the nominative case would be Мы все рады - We are all happy.
I see that you study German, so it's sort of the same principle as "Es ist mir kalt" (in Russian it's actually the same - мне холодно).