What does the -p3 option to the patch command do?

It will strip off path information from each file mentioned in the patch file up to and including the third / character.

patch continues execution as long as there are three or fewer errors.

It instructs patch to look up to three lines of context before or after the declared line in the original file for a match.

patch will keep three previous versions of each file in the output to prevent loss of change history.

It instructs patch to conform more strictly to the POSIX standard.