It means that they appreciate fine foods.

Hi Hisako, You asked a very good question! Well, what does this phrase, "sophisticated palate"mean? Let's review these two words closely. The word sophisticated refers to a clever, modern (up-to-date), well cultured view or approach. The word palate, unfortunately, has more than one meaning. However, it is understood that in this sentence it refers to taste/preference. (None of the other meanings make sense due to the context) Ok. In summary, the phrase sophisticated palate means a "modern, well cultured preference" K.C. Feel free to contact me if you have any other questions?