The attack on Donnager: why are the attackers risking their lives for this mission?

If you have only watched the first (few) episode(s), I suggest you don't read this answer and watch the rest of the episodes.

I don't quite understand the premise of the question. Private military contractors and mercenaries (so-called "soldiers of fortune") have existed for a considerable amount of time in real life. Why do people become a mercenary and put their lives on the line? Money.

When escaping from the Donnager, Holden encounters one of them. The enemy soldier replies: "I am dead already". What does he mean with this?

This suggests that the enemy soldier is working for someone who will kill them if they fail to accomplish their mission. This is not too dissimilar from e.g. how mob families are portrayed, where interacting with the enemy (the police) is liable to get you killed by your own mob family.

"I am dead already" means "You (Holden) can't threaten me with death. Even if not by your hand, someone/something else will soon kill me."

What are they exactly? Mercenaries? Fanatics? Brain-washed high-tech soldiers?

This hasn't been expanded on (in the TV show at least, can't speak to the books).

I don't know how far you've gotten in the show (since you posted a pilot episode question) so I'll avoid spoilers, but much of the concrete workings of who sent the ships has never been explained. The show does explicitly state and suggest that they have a very wide reach and deep pockets.

The most likely answer is money, but in later seasons, you will also encounter characters who have aligned with the "evil" (for lack of a better word) side because they genuinely believe it is necessary to do what they're doing.