The mean of a group of numbers marks the "balance point" of the whole set, the single value everything is centered around.
If every observation is nudged up by the exact same amount, the entire block of numbers slides upward together without changing how they're spread out relative to one another.
Since the balance point of a group moves exactly as far as the whole group moves, shifting every value by 5 shifts that balance point by 5 as well, no more and no less.
This differs from operations like multiplying every value by a constant, which would scale the mean rather than just shift it; here we're only adding, so the change is a simple, direct shift.
So the correct answer is increase by 5.