If you believe that the mind is divided into the conscious and the unconscious, then it is possible to explain this phenomenon. Your conscious mind is unable to process enough chunks of information to solve the problem (presumably complex), so you need to allow your unconscious mind to do the processing.
The problem is, by its very nature it needs to process information unconsciously, which means that if the conscious mind is processing the information at the same time, the efficacy of the unconscious mind is compromised.
This explains why we need to "let go" of the problem and all its associated information and work on other things. By doing so the information is available only to the unconscious, and it is therefore able to process it more efficiently.
I have many instances where I have difficulty resolving an issue, for example trying to understand a mathematical concept, and I constantly obssess about it, to no avail. Sooner or later I had to let go, and so I did. After leaving it alone for a few days, I came back to re-examine the concept, only to find out that I understand it completely!