It's not my intention to spread the use of slang on this forum. I've never seen the phrase either, well, until 1 hour ago when I read it in an article on tvtropes.org. Here is the link:
"Scrub" was originally (Late Middle English) just a variant of "shrub". The idea of being dwarfish and stunted as compared to a tree apparently led to the various derogatory meanings the word acquired. It has been used to mean an inferior person since the sixteenth century.
There are a few uses in my dialect, such as an appellation for small varieties of trees like the scrub pine. I've he
A scrub is a generally disagreeable person who tends to aggrandize himself, his friends, or his cause at the cost of "common" civility. A scrub doesn't necessarily hang out of the passenger side of his best friend's ride, but he may. The primary marker is a disregard for others' feelings.
From the UK, I would say that a 'scrub' is someone whose name has been scrubbed from a chalkboard
For example, bookmakers used to chalk up odds on a board but then 'scrub' the horse/rider if it was a non-runner
I think it means that the person is no longer considered to be 'in play'. Our US colleagues can advise whether it means something like 'put on the bench'