What does it mean - they are running a beam? (Its about a raft caught in the waves)?

abeam, astern, ahead are three orientations relative to the craft.
NOT a beam, a stern, or a head.

Ahead is coming from in front.
Astern is coming from behind
and Abeam is coming from the side of the craft.

A lesser used term these days is abaft.
but as aft is the rear you should guess what it means.

They were going sideways down the river/waves

Two boats running abeam means they are running side by side with some distance between them.

The "beam" is across the width of the raft. So running abeam (one word) means the raft is being pushed sideways by the waves. In general that's not good because the raft can be more easily upended when being pushed sideways.

the word is abeam, which is:
On the beam, a relative bearing at right angles to the ship's keel.

in this case, the waves are at right angle to the keel, coming from the side.