Generally, as a man I "shut down" in an argument under several circumstances. It might be difficult to tell which, but in any case your best bet is to let it go, and resume talking when he is ready...
Here are the common reasons that cause me to shut down during an argument. There are more, but this is a good start.
1. Whatever you just said defies all reason and logic. It's going to take me a considerable amount of time to decipher what the hell you even said, let alone what you meant. The logic of women defy normal reason sometimes, and when you attempt to justify your argument based on female intuition, male logic just can't cope with it. It will take time to sort this out and come to a conclusion. Once we figure it out, we can continue our discussion if it is still an issue that needs to be resolved.
2. You are wrong, and there is nothing I can say that will convince you that you are. This argument will last forever if one of us doesn't let it go, and I for one choose to let it go. Perhaps I am trying to formulate a new response that will help you see how wrong you are, but perhaps I have given up trying to be reasonable with you.
3. I am wrong. Something you just said has given me pause, or has convinced me that perhaps I have not thought this through. It has made me think and reflect on myself, and I am now going over the argument step by step to figure out how much of an apology I owe you, and what I need to start doing different. An apology is coming... so just be patient while I work out how much damage I have caused.
4. I'm just tired of arguing and do not want to participate. This is the same argument we had yesterday or last week, and you are not offering anything new to the subject. You are just re-hashing the same old bullshit. Leave me alone, and don't bring this up again until you have something new and helpful to contribute to the argument.
5. I am hurt and offended by what you have said or accused me of. There may or may not be elements of truth to it, but regardless it hurts me deeply that you feel this way, and as a result I am shutting down as an emotional defense against being hurt more. Men generally do not like to be perceived as vulnerable, but make no mistake, we are. We damned sure don't want to show that weakness, especially when you are being cruel to us. Chances are, you owe me an apology, and I will likely shut down on this subject permanently until you recognize that, and apologize.