The expression “to cut the mustard” means: To accomplish, be able to, or succeed with; to meet expectations; to play, as music, expertly.
To get at the origin of this altogether American expression we have to go back to the beginning of the century when “to be the proper mustard” was a slang phrase meaning to be the genuine article, possibly because some so-called “mustard” of that period would not pass today’s pure food requirements.
From that, immediately, came “all to the mustard,” that is, all one could ask for, fine and dandy, “copesetic,” as the late Bill Robinson would say. Then hotly, as early as 1907, came our present phrase.
O. Henry used it in Heart of the West in that year: “I looked around and found a proposition that exactly cut the mustard.”
It’s just a slang expression, bearing no connection to the use of the verb “cut” in the sense of to reduce the strength of.