I can, if you're willing to you're willing to consider the response carefully.
(First off, who cares? Nihilism is, in my opinion, a mistake. It's a nice idea in theory, but fails to take into account human psychology and the need for 'illusions'. Sure, go through a period of 'freedom' in order to 'choose' your battles, but this, like like Sartre's existentialism, assumes that after you violently dislocate motivation from its point of contact with 'reality' (i.e. after you go through the process of nihilism, or 'cleansing the doors of perception' to see the Void) motivation just doesn't wither and die. In other words, if you dwell on the fact of the matter than objective meaning is an illusion, it might be quite difficult to ever give a shit about anything ever again. For, how can you simply forget the fact that objective meaning is an illusion after you have realised this? That would require a process of 'getting dumber' again after you have 'gotten smarter'. Also, some people who i've spoken to around here seem to think that after you learn the facts that deconstruct objective meaning, you will suddenly be able to see that certain values are somehow objective (authoritarianism, paganism, perennialisn, deep ecology, or whatever), while other values are somehow not (liberalism, christianity, humanism). If nihilism is true, no values are binding whatsoever. Certain values may preserve the natural world better than others, for instance, but so what? It is a further leap from this to the conclusion that you SHOULD follow them.)
To the question. There are two meanings of the word 'meaning'. On the first, 'meaning', is just the way in which a language represents/refers to the things it does. The meaning of 'cat' is the cat over there. The meaning of 'happiness' is a certain mood grounded by certain neurological processes. The cat being happy means the cat over there is in a neurological state. Obviously even a nihilist accepts this sort of meaning. His words still refer to things due to being involved in a language community which has conventions regarding what words refer to/represent/mean. 'Cat' in my language convention that i share with other speakers of English means a certain mammal with four legs that has been bred from other mammals over a long period of time to chase mice and balls of string and shit on your carpet while only being concerned with your existence when it is time for dinner. 'Cat' (i.e. the set of symbols c-a-t) in the language convention that speakers of French share means nothing - it picks out or refers to no object in reality (I know not what set of symbols in French picks out the appropriate mammal).
On the second meaning of 'meaning', meaning means(!) purpose. Then what does purpose mean? Purpose means some prior intention informing an activity. You moving your arm to pick up the beer is an activity informed by a prior intention (your intention in drinking the beer). A nihilist uses the conventions of his language ('meaning' in the first sense) to convey the idea that there is no purpose informing his activities other than his own wishes and desires. Similarly, he uses language to convey the idea that there is no purpose informing the activities of the universe because those activities are not caused by an agent, and only activities that are caused by agents can have purpose.