Everyone has multiple dreams per night. One reason that we don't remember them is that we place so little significance in dreams. However, some African/Pacific Island cultures believe that dreams are a form of prophecy and/or communication with the dead, and they have trained themselves to recall dreams. They do such with vivid detail and great accuracy.
If you are woken up during your REM stage (which is the dream stage,) you tend to be able to grasp at dreams and remember if you make an effort. Since the amount of time spent in REM sleep increases proportionally as the sleep cycle goes on, then your new sleep habits may just mean that you are in the dreaming stage more.
EDITED to add - Dreams occur in real time, if you didn't know. You can't have "thousands" of dreams in the common usage of the word, only perhaps thousands of random brain-activating, dream-inducing neuron firings.
You are probably remembering more dreams because of the new sleep cycle that you have adopted. Or perhaps you are just becoming more aware of the dreams, which in turn make you remember even more. My psych prof never used to remember his dreams. He actually attended a dream seminar, in which they gave him strategies, methods, and training to remember dreams. Now he claims to remember 4-5 a night, which is the highest rate I have ever heard of.