Learn German: Du läufst über das Brett und schwingst über das Wasser hinüber. - You run over the board and swing over the water.

It acts as a separable prefix for the verb schwingen, forming the compound verb hinüberschwingen (to swing across). In a German main clause, the conjugated verb (schwingst) stays in the second position, and the separable prefix (hinüber) gets kicked to the very end of the sentence.

What is the difference between über das Wasser and hinüber? Doesn't that mean 'over' twice?

Yes, it sounds slightly repetitive to an English speaker! German frequently pairs a prepositional phrase (über das Wasser) with a matching directional adverb (hinüber) to clearly emphasize the completion of the movement from one side to the other. hinüber specifically means 'over to the other side, away from the speaker.'

Could I use herüber instead of hinüber here?

Only if you change your perspective! You use hinüber when the person is moving away from you. If you were already on the opposite side of the water and the person was swinging towards you, you would say herüber.

Why do we use the accusative case (das Brett, das Wasser) after über?

The word über is a two-way preposition. Because the sentence describes an active change of location (running across a board, swinging across a body of water), it requires the accusative case. If an action has no destination and you are just staying in one place, two-way prepositions take the dative case instead.