This phrase describes the act of one person using the status, accomplishments or position of another person to further their own personal gains. It can also be taking misdirected credit where none is deserved.
For example, naming your website www.everything3.com with the intention of catching all the people who make typos. Another good example would be someone who excessively uses their bosses name/authority at work so that their projects, or directives carry more weight. Also, taking credit for a solution to a problem at work because you are the point-person for the problem, when your co-worker came up with the solution. That example is quite common.
There are varying degrees of borrowed glory, but they all seem to be "sleezy" and their acceptance is based on the situation and circumstance.