I think that this is a poorly chosen question in its formulation as it starts with the "incorrect" version using the possessive case and provides the answer with an adjective replacement for the possessive. I think that both our class' picnic and our class picnic are acceptable alternatives, unless there is some other condition applying to the exercise which would limit the answer to one or the other.
With respect to the possessive, while there are various grammar rules for words ending with 's' which differ from region to region (as here, class's vs class') either usage is generally accepted.
Consider if the sentence were, "We all looked forward to going on Bob's picnic." Bob is not possible as an adjective so it can't be changed but if it were, "We all looked forward to going on Tess's picnic." then it could also be written as, "We all looked forward to going on Tess' picnic."
The difference is that class can be used both as an adjective and as a possessive pronoun: our class's rules and our class rules both mean the rules of our class.