Very rich, possibly having become so by unfair means.
This little phrase can't be explained without looking at the word lucre. From the 14th century lucre has meant money and is referred to as such by no less writers than Chaucer and John Wyclif. These references generally included a negative connotation and gave rise to the terms "foul lucre" and "filthy lucre", which have been in use since the 16th century. "Filthy lucre" appears first in print in 1526 in the works of William Tindale:
"Teachinge thinges which they ought not, because of filthy lucre."
Tindale was here using the term to mean dishonourable gain.
Following on the the term "filthy lucre", money became known by the slang term "the filthy", and it isn't a great leap from there to the rich being called the "filthy rich". This was first used as a noun phrase meaning "rich people; who have become so by dishonourable means". It was used that way in America, where it was coined, from the 1920s onwards. Here's an item from the Ohio newspaper The Lima News, February 1929, which deplores the get-rich-quick attitudes of some who were exploiting those who had to sell their homes at unreasonably low prices in order to eat during the economic crash:
"There is a depressed market. If any of our stock-gambling filthy rich want a winter home, now is the time to acquire it."
As time went on the negative associations have softened somewhat. It has become to mean "extremely rich" rather than "dishonourably rich", although there may still be a trace of an unfavourable implication associated with it.
See other phrases that were coined in the USA.