It is correct English. It is a common word.
It literally refers to farming or gardening. It means to take care of plants, or gardens, or crops. Sometimes it means specifically to prepare the soil, by loosening it, so that plants will grow well in it. We "cultivate" plants in order to make them grow.
We also speak figuratively of "cultivating" a skill or an ability. Again, it means taking care of it and making it grow, usually over a long period of time. "He tried to cultivate a sense of honesty in his children." "She went to art school to cultivate her artistic ability." I see that an ad from a school that teaches rock music says "cultivate your inner rockstar."