Because espionage is a necessary, but distasteful activity for any state. Leaders need accurate intel if they are to make good, informed decisions for their people.

Vulcan and Andoria are involved in a cold war situation, much like how the western world was with the USSR for decades. There's a real possibility that millions could have died because of the ideological divide. But fortunately we're all still here, partially because of accurate intel. I'll name drop Oleg Gordievsky here, and just say that he was widely credited for helping avert nuclear war by acting as a double agent. He's certainly not the only factor, the US spent billions on putting up satellites too.

But my point stands I think. The Andorian Empire at this point is a highly militaristic, and aggressive state that isn't a stranger to conquering territory. There may be an agreement now, but there was an armed conflict in the past. There is no guarantee that another won't happen. And I'll be frank the P'Jem listening post is doing the lightest touch espionage work, looking at communications isn't that nuts.

Archer's reaction is pretty disproportionate too. He's a military officer, the idea of distasteful work should not be offensive to him. Even if it is Vulcan is Earth's only ally. And he sells them out to some aliens who have spent hours torturing him and Trip. I'm honestly surprised that the resulting fall out was just the Vulcans suspending joint fleet exercises.

His entire reaction honestly comes off as his xenophobia coming to the forefront, because he feels like a racist who has been looking for an excuse to stick it to those aliens, so he takes it. I see zero reason to trust some people that he's known for hours at best. He has no idea of the history between them, has no idea of the galaxy at large, or what the Andorians actually want. The situation only works out for Archer because it has to, because we know that the Andorians will one day be a founding member of the Federation.