Does cosmology answer why the universe exist?
Or only "how"?
I don't believe that it can even explain 'how'. (There are hypotheses though.)
'Why' is something else. It seems to suggest that the universe exists as the result of a human-style
act and that the act needs a
motive,
purpose or
goal.
That idea comes about through our very human tendency to understand events as if they were part of a larger
narrative. It's the tendency to interpret reality as if it was a literary story. Reality receives a
plot.
So we get religious stories about the act of creation and about the purpose and ultimate end of creation. People like to think that way because they can plug the seemingly meaningless events of their own lives into the plot of the cosmic story so as to give the events meaning.
It's how we understand social life and the actions of others, after all. (And that might be part of why humans evolved to think that way.)
What is the meaning of life in the eyes of a cosmologist?
Physical science is similar in that it plugs particular events into a larger narrative which allows scientists to make sense of the event. But now the narrative is the narrative of a textbook and not a novel. So a physical event is plugged into the narrative of there being 'laws of physics' which explain what happened and make it more comprehensible. Facts about lifeforms are plugged into the evolution narrative (like I did up above).
But the narratives of science are far more impersonal and abstract than the narratives of social life, literary novels or religion. They are the plot device patterns of physical reality kind of reduced to pure mathematical form (in theoretical physics' case at least).
So to answer the question, in the eyes of the cosmologist meaning is replaced by mechanism, so to speak. The cosmologist explains an astronomical discovery by plugging it into his principles of cosmology that tell him how it might have come about and what kind of initial conditions might have led to it. The cosmologist doesn't interpret the events by relating them to some more humanistic cosmic story about goals and purposes that the astronomical events might have illustrated in the heavens.