Isn't "I don't have time to respond" an oxymoron?
I have read in an answer that professors are also human beings. Exactly, and decent human beings don't leave others so anxious and lost with a response to the point that they have to go ask in a public forum. Also, talking about human beings, the professor would never give this answer to a peer or someone ranked higher than him.
I don't find that response appropriate or excusable. It would have taken literally 30 seconds to add to that response what the next step is. For example "I will get back to you in 2 months", "I am sorry but currently I am not accepting more students", or "meet me in my office in 2 weeks".
So, no one really knows what your professor means, but if I had to bet, I would bet he is not interested in you at the moment but he does not want to appear impolite or inconsiderate, so he thinks this is a better answer than answering honestly. And this is like in relationships. Please, don't insist and beg, or he will smell your desperation and will either feel annoyed/stalked or get you and exploit you. If he is interested he'll get back to you. In the meantime, look for someone else who can spend 10 seconds writing a proper answer.
Regarding being busy and what "I don't have time" means, I recommend you read this article, because it's answered there: http://scottberkun.com/2010/the-cult-of-busy/
I will quote directly here, for those busy people who don't have time but are reading Academia StackExchange anyway :)
The phrase “I don’t have time for” should never be said. We all get the same amount of time every day. If you can’t do something it’s not about the quantity of time. It’s really about how important the task is to you. I’m sure if you were having a heart attack, you’d magically find time to go to the hospital. That time would come from something else you’d planned to do, but now seems less important. This is how time works all the time. What people really mean when they say “I don’t have time” is this thing is not important enough to earn my time. It’s a polite way to tell people they’re not worth your time.