"Shabbat" is the Hebrew word for "sabbath." "Shalom" is a greeting that works as both "hello" and "goodbye." Hence, "Shabbat shalom" is a greeting for the Sabbath. Israelis colloquially use it on Thursdays and Fridays as a rough equivalent to the American "have a good weekend."
As a side note, "shalom" literally means "peace." Thus the person saying "Shabbat shalom" is wishing you a peaceful Sabbath.
As a side note, "shalom" literally means "peace." Thus the person saying "Shabbat shalom" is wishing you a peaceful Sabbath.