The primary reason to do a palace jump is make a different area other than your starting area into your primary productive core.
For instance, if you start off on a small island or a peninsula, you can't surround your capital with many cities. Because for non-communistic governments, corruption is partially due to distance from the capital (or FP), you want to have a large number of cities spaced around the capital to keep the core from becoming too far-flung. Thus, consider moving the palace to a more central location.
Or, perhaps your starting area is resource-poor and you want a higher-potential area to become the core.
You can jump the palace a few different ways:
1. Use an MGL to rush a palace build in a new city (the palace counts as a small wonder)
2. Abandon your original capital city and the palace will flip to a new city. That new city is chosen based on a formula influenced by city size, military garrison, and possibly a few other things. Like Tribute said, CivAssist tells what city that will be.
3. Simply naturally build the palace in a new city. How many shields it takes is based on the number of cities in your empire.
4. If enemies capture your capital, the palace jumps like in #2. Not a recommended option.
5. Use an SGL to rush the palace, like in #1. Also not usually recommended because SGL's should be used for great wonders.
Using #1 or #2 is preferred if the new capital will be in a highly corrupt area because the natural build will take too long.