Just to add to what other members have kindly contributed, I think this might help you out, because it can become a little confusing:
"irme" does usually mean "leave":
"Quiero irme de esta casa." ( I want to leave this house. )
"Voy a irme de aquí." ( I'm going to leave this place. )
The difference with the construction you're asking about, is that it's like a combination:
"irme" + "de viaje"
If you were to translate it literally, you could say it like this (although this is not usually how you end up translating it):
"I'm leaving on a trip." or "I'm leaving to go on a trip."
So, it's two ideas put together, you're leaving, but you're leaving because you're taking a trip.
That's why this specific construction in Spanish, that is "irme" + "de" + "viaje", the right way to translate it is:
"I'm going on a trip." or "I'm taking a trip."
Buena suerte! Hasta la próxima.