Why would one make the switch from extract to partial mash?
Here are a few reasons some choose to make the switch from extract to brewing partial mash:
To learn mashing, sparging, and lautering techniques on a small scale.
To add a "fresher" malt flavor to their beer (this could be debated).
To use grains that must be mashed, for which there is no extract available.
To feel a bit more connected to the brewing chemical processes, in that it requires steps that are not employed using only LME.
It is a bit more work and lengthens the brew day compared to extract brewing, but this factor can be balanced out by the paired cost savings, base malts are much less expensive than extract. (Plus, if you enjoy brewing, you get to spend more time brewing!)
Grain insight into the different characteristics given by similar malts from different maltsters or countries. There are wide varieties of malts to play with. Part of the fun is to try them out to find your favorites for use in your own recipes.
What is a partial mash? PDF