"What do Koreans mean by the word '센스'?"
When we say "sense" without a qualification, as in "센스가 있다", it is used in a broad and slightly vague sense(^^). It includes a sharp perception, quick wit and sensitivity (in things interesting and in vogue), good judgment and understanding attitude - the opposite of dull, dense, uptight, clueless...
It depends on the speaker too, as it is likable qualities judged in one's value system.
Among young women for example, it might be more of a fashion sense, chicness, and chivalry, while for men it would be decent sensibilities, big-heartedness, etc.
It also depends on the context and how it is said.
So 센스가 있다 is a strong compliment indicating all the good qualities.
센스가 빠르다 on the other hand refers to quick (and sometimes shrewd) perception and adaptability, so somewhat less of a compliment.
~에[의] 센스가 있다 is close to "has good sense in.." in English.
센스 있는, 센스 있게 means perceptively, sensibly, with keen perception/insight, etc.