As the other answer has described, this is sarcasm, but it doesn't really explain the meaning. First, eulogy has two meanings, it also means a praise about a person recently deceased (an obituary is only a notice of death, with a short biography and maybe facts about their life while a eulogy is a tribute to the late person). But it can also mean a tribute to people still living. So:
Somebody only really gets such praise in two cases. First, if they are wealthy and powerful. In this case, the eulogy is implied to be mendaciously flattering in order to gain favor from the important person.
Second, if a person, not really liked and loved by their fellow people, finally drops dead, so that all the others can finally have their joy. This eulogy is just as mendacious as the first one, as there was nothing to be praised in the first place. The only thing worth praising about this fellow is that they're now dead.
So, in the end, as Bierce posits it with a dark twist: all eulogies are lies. If the recipient is still alive, we only make the eulogy to please them, not because it's true. And if they're dead, we only make the eulogy to mask the fact that we didn't care about them, but we're happy to see them go finally.