If part of the reward is cargo, then you will need to have cargo space available on your ship.
So far, I've seen two possible explanations, both of which are documented:
In either case, you can't complete the mission straight away.
- The mission is bugged. This sometimes happens if the destination is updated en route. It takes a wee while for the updated mission to show at the target station.
- You don't have enough cargo space to store part of the mission reward, which is cargo.