Learn Finnish: Surffaajat kantoivat surffilaudan rannalle. - The surfers carried surfboards to the beach.

This is an example of the distributive singular. Because each surfer in the group is assumed to be carrying exactly one surfboard of their own, Finnish prefers using the singular form. It conceptually distributes one board to each person in the plural subject.

If I used the plural surffilaudat instead, how would it change the meaning?

Using the plural surffilaudat would imply that the surfers as a collective group carried a pile of surfboards together, or perhaps that each surfer carried multiple boards. The singular clearly flags that it is a 'one each' situation.

Why does surffilaudan end in -n (genitive-accusative) rather than being in the partitive case?

The action of carrying has a clear, specific destination (rannalle), which makes it a completed, goal-oriented action. Therefore, the object takes the total object form (the genitive-accusative) rather than the partitive, which we would use if they were just carrying boards around aimlessly or if the action was incomplete.

Why do we use the allative case rannalle (onto the beach) instead of the illative rantaan (into the beach)?

A beach is treated as an open surface in Finnish, so we use the external local cases to talk about moving onto it. Rantaan is also grammatically correct but has a slightly different nuance; it usually implies going right down to the water's edge or arriving at the shore from a boat, whereas rannalle is the standard way to say you are going to the beach to spend time there.