To garner means to gather or earn. If you want to run for office without belonging to a political party, you must garner enough signatures — usually a few thousand — to get onto the ballot.
The word garner comes from the Latin granarium which means "store-house," usually for grain. The current use of the word carries with it the sense of something being stored up. It’s not only that you can gather enough yes-votes to overcome the opposition, there is the sense that all the things you have garnered have some weight of their own. In its other use, garner means “earn” or "merit." Think of movie stars garnering Oscar nominations for their excellent work.