First of all, try not to be so shy in matters like this. There is usually someone around you can ask in a discrete fashion, and not only will you become wiser, but you will also gain greater respect from the person for your wanting to learn more.
Many people have a custom to say the words “Lichvod Shabbat Kodesh” — “In honor of the holy Sabbath” —before eating each course. This is to show that the eating is not for purely physical reasons, but also because it is a mitzvah to delight in the three special meals that are served on Shabbat. Saying this aloud (quietly is fine, and in any language) is also an affirmation that one is acting in accordance with the command to “Honor the Sabbath day”, which is one of the Ten Commandments.
I must admit that I although I have been in homes where people do as you heard and saw, most homes I have visited do not make this statement before each course. Some say it before the first course, and some do not say it at all. I think that mainly certain groups have this custom, and I have seen it in Chassidic and “multi-generational” Jerusalem families. However, even in homes where it is not explicit, it is implicit. The comments of gratitude and “special-ness” made by the participants, and the general special Shabbat “atmosphere” at the table speaks for itself. Eating the special foods with the special “spice” of this day is done for more than mere physical nourishment — it is also being done to honor Shabbat.
