The Licensed Professional Exemption is basically a streamlined documentation path that is available for certain credits, particularly design-oriented credits where a licensed professional such as a Landscape Architect or a P.E. would be in a position to judge whether the project is compliance. In most credits this option is not available due to the nature of the credit, e.g. an MR credit that requires actual tracking of purchases.
It basically involves substituting the signature of  a professional for some portion of the usual documentation. This is shown in LEED Online in the relevant credits.
Unfortunately, USGBC or GBCI have not released much information on the LPE path, such as a list of the credits where it's an option, and general guidance on using the exemption. There is a small amount of info on it in the LEED Reference Guide.
The benefit of the LPE path is obvious—efficiency. The downside is that in order to feel confidence signing off on a credit, the professional may need to develop all the documentation that LEED would require, anyway, and instead of putting their professional authority on the line, they might as well simply submit the documentation.