How could he know that he really doesn't know anything ? Did he check everything to rule it out ? I think he should have said that he "Believe" he doesn't know anything and by that he would also avoid the paradox of knowledge and state that the only thing that exists is belief and not knowledge.
In my opinion, you should understand it as admitting we can't know anything for sure. No matter how confident we are about the validity of some statements, theorems etc., we can never reach 100% certainty. You might say, well, why do we generally believe things are not certain instead of believing in them with no criticism or doubts - I guess that's just the way we were 'built'.
We should ask another question - why me, you and most, if not all, other people tend to think confidence about validity of theorems in any science (biology, physics, and even mathematics that we may consider unfalsifiable) is unreachable? And then, why you asked this question even if you feel you can keep asking such questions forever, that's why you will never get the ultimate answer (and then ask the question if it's true there are no ultimate answers).
I like the interpretation of this position given by Sextus Empiricus: That the best approach to knowledge is to continually set aside one's concern for whether or not something is true. Thus it is not just belief replacing certainty, it is continual avoidance of belief, and seeking to hold all opinions in appropriate perspective.
I think it is likely to be what Socrates and the Cynics had in mind when they said the wisest man was the one who knew nothing, because believing something wrong was worse than being ignorant.
Of course, this, like many other Golden Mean interpretations is a self-contradictory position, if you believe it. Moderation in all things but moderation, tolerance of everything but intolerance, etc. But it escapes this paradox in practice, because if you apply it to itself, you don't really believe it, and it does not bother you to act upon it.
user107986's said that Socrates was suggesting that we cannot have certainty, and this is a possible reading. I'm not sure I agree that this is what Socrates had in mind, but it is a worth-while consideration; and whether or not that's what Socrates was saying I do think it's a good thing to think about anyway. It certainly is an answer to your question, since it demonstrates one way in which Socrates wouldn't have to check everything in the world to see whether he knows none of it.
I would suggest, however, that another reading of Socrates is that he means to tell us some mixture of the following two things: One, there is nothing he can identify, which he can claim to know it. This is subtly different from saying that there is nothing that he knows. Two, he may also be indicating that we don't even know what knowledge is--we do not have a definition of it, although in one dialogue he may eventually accept that justified true belief defines knowledge. After that point, it seems that perhaps one thing he knows is what knowledge is. But before this point, he might have meant to say that without knowing what knowledge is, then you cannot be justified in claiming knowledge of anything.
Also, especially when dealing with the ancients, if you could take their claims and replace a single nearly-synonymous word to make the claim more sensible, you should probably assume that this was or could have been the person's original intent. After all, you're getting a translation of a language that is no longer even in use (ancient Greek is sufficiently dissimilar from modern Greek that we're not even completely sure how to translate certain loaded words like knowledge or belief). So the English that you're looking at is, at best, an approximation.