Is it possible to have evidence that something exists without having an explanation for why the thing exists?
Yes, it is possible and reason for it depends on the temporal nature of existence and "why".
To empirically find that something exist would require a less temporal scale as compared to find why that something exist. The question of why can go on way too far in the past or way too far in the future. So basically you can find the existence in very small time scale whereas "why" would require much larger time scale to get answered.
For example: You could come to know the existence of something on your table as soon as you look but to reason why it exist would require you to spend more time as compared to "just looking at it". For some moments you will know the existence of the object on the table without knowing why.
Take the cosmos: I assume that we all agree on its existence. But no generally accepted answer is known to the question Why does the cosmos exists?
Enclothed in philosophical terms it is Leibniz' question Why is there something rather than nothing? But one can also doubt whether that is a reasonable question.
I'll answer a broad question in a broad way, and say yes, you can have evidence that a thing exists without knowing why it exists.
To argue otherwise it's necessary to make a couple of assertions, which must both hold true:
- Every single thing that exists has a purpose behind its existence; purposeless things categorically do not exist.
- The purpose behind any thing in existence is something that will be readily apparent and comprehensible to human intellect.
The first assertion appears to be an appeal to the notion that there must be a divine intelligence which has created and ordered the entire universe towards some ultimate purpose (as how else can you get an entire universe in which not even a single atom is misplaced or "purposeless"?).
That's very difficult to prove, in and of itself. And also infinitely recursive, because if such a divine intelligence exists that intelligence itself must have its own purpose for existing, in an even larger context. And that larger context must also have its own broader purpose.
And so on. If any context exists without a discernable purpose and a larger context around it, the original premise is disproven as there would then exist at least one thing which has no purpose.
And the second assertion is no less problematic. It essentially requires that human beings be omniscient, able to divine the true purpose of anything they might observe at the moment they observe it. Which is manifestly untrue. Though even slight untruth is sufficient to disprove the premise that anything that exists must have a known/knowable purpose for existing.
One could argue that perhaps everything does have a purpose, but without human omniscience that purpose may be so deeply buried that mankind has no chance of ever finding it. Or so blue and orange that we'd never comprehend the purpose even if we found it. Which still fails the original premise.