Keeping Yom Tov like keeping Shabbos
My grand father asks what is the conceptual connection between the two? He offers two answers:
One is that as Shabbos is associated with recognizing yetzias Mitzrayim,[see Ve'etchanan 5:15]as we say in it is a fundamenal foundtion for all the holiday, and so the two are inextricably linked.
The other reason is that keeping Yom Tov is a sign that one keeps Shabbos for the mitzvah and not just for a weekly day of leisure. As Yom Tov tends to fall on weekdays, they are a break in the standard division of workdays and rest days. They are outside of the standard routine one sets up for the week and so keeping them indicate that one is not acting for one's own preferences but for the mitzvah. In that way, the observance of Yom Tov attests to the observance on Shabbos as also being done for the mitzvah.