When a surprising new result comes out, we often try to understand what happened: What is new about this? How come we could not solve the problem earlier? Then a pattern appears. Often, exciting new results happen by making a connection between the problem at hand and a seemingly unrelated field. Connections between a problem and other fields can work magic, and gives insight into the hidden meaning of mathematical phenomena. Researchers who are transfers from a different field can have a great impact in that way, since they can "see" structure that is invisible to other, more mainstream researchers.
How can we hope to make such connections? A broad mathematical education is crucial for that. Knowing, not only the standard tools of our trade, but also results and techniques from related fields and even from unrelated fields (if one hopes to get lucky). That's why, before students start doing research and while they are doing research in the course of their PhD, they take courses not just in their area of specialty but also in other areas. In fact, in some departments I have visited (Berkeley, U. Washington,...), the faculty also sit on one another's lectures to try to broaden their knowledge. Mathematical culture enables one to find more meaning in casual remarks so as to develop them to create theorems.
If I don't have it, I don't know what I am missing. Then, at best, I prove results 'by hand' without being aware of the larger frame into which they fit.
Similarly culture enables one to find more meaning in life; but if I don't have it, I don't know what I am missing. Then I think that, say, museums are boring, and I cannot understand what people find in them.