Skip to the bolded part of the last paragraph for the tl;dr version of this answer.
There's a trope in fiction where characters whose identities are in doubt (i.e. they might be an imposter, or possessed by someone else) are asked to prove they are who they say they are by saying something only they would say, or know.
For example, in X-Men (2000), there's a scene where Cyclops suspects Wolverine might be Mystique in disguise, so he asks him to prove he's really Wolverine, and Wolverine does so by telling Cyclops that he's a dick. This wasn't much in the way of proof, but it was sufficient for Cyclops in that context, since Mystique likely wouldn't have known of his and Wolverine's dislike for one another.
In this case, 'Darkhold Strange' asked 'our' Strange to prove he was who he claimed to be (i.e. another Strange from another universe), and Strange reasoned that one way of doing so would be to share a childhood memory that only another Strange who experienced a similar childhood would be likely to know.
DARKHOLD STRANGE: Who are you... what are you?
STRANGE: I'm just, one of us.
DARKHOLD STRANGE: From the Multiverse?
STRANGE: That's right.
DARKHOLD STRANGE: Prove it.
STRANGE: We had a sister. Donna. She, uh... She died, when we were kids.
DARKHOLD STRANGE: How?
STRANGE: We were... playing on a frozen lake, and... She fell through the ice. I couldn't save her.
DARKHOLD STRANGE: Sounds about right. But we don't talk about that, do we?
STRANGE: No, we don't.
There's no guarantee that the sharing of this memory would've been sufficient, as Darkhold Strange might've had a very different childhood for all our Strange knew. But Darkhold Strange's reply suggests he did experience that event also, and that this revelation did therefore persuade him that our Strange was telling the truth.
The sharing of this memory also doesn't directly prove that our Strange was from another universe, but as indicated by Darkhold Strange's dialogue, he already had that possibility in mind, and in having been persuaded that he was speaking to another Strange, it made sense to also conclude that this other Strange was from another universe.
There are other possible conclusions -- like this other 'Strange' could've been an imposter with the ability to read minds -- but I think the point of this exchange was not for him to prove beyond all reasonable doubt that he was another Strange from another universe, but merely that on the balance of probabilities, this was the most likely explanation.