On Shabbat and Yom Tov, usually one person makes ha'motzi and everyone else at the table says amen and eats some bread. As far as I am aware, during the week, nobody ever eats food without making their own bracha. Why is simply responding "amen" permitted for shabbos and yom tov meals?
Short answer: They may want to rely on the brocha of the master of the house on Shabbos in order to be included in his lechem mishna, although this may not be necessary,
There are two types of brochos: 1) Birkos Hamitzvos; recited (usually) before performing a mitzva 2) Birkos Hanehenin; recited before and after having pleasure from something (such as food). While by Birkos Hamitzvos, it is sufficient to be obligated in the mitzvah to be able to recite the brochah on behalf of another, this is not the case by Birkos Hanehenin, where the person deriving the benefit must personally say the brocha. In order to say the brocha for another, the pair must be be established as eating together at the same table to be considered one unit. (Based on Shulchan Aruch Harav 167:17)
In a case where people are sitting together at a table to eat and fulfill the conditions needed to be included with each other, it would in fact be better for one person to say the brocha on behalf of the other participants, fulfilling "ברוב עם הדרת מלך". However, due to concerns that people are not careful about avoiding an interruption between the hearing of the Brocha and eating the bread, the custom nowadays is for each person to make his own brocha. (ibid. :18)
On Shabbos, there is an obligation to have lechem mishna; two loaves of bread. In a case where lechem mishna is not available for each person, the leader of the house will make hamotzie on lechem mishna on behalf of all present. (See Shulchan Aruch Harav 274:4). Some mantain that in order to be included in this they would also have to be part of his brochah homotzie.
However, the Lubavitcher Rebbe writes in a letter (Iggres Kodesh Chelek 14 Page 29, see here) challenging this, and asserts being included in the hamotzie brocha is not a precondition to being included in the lechem mishna. In fact, one is able to be yotzei lechem mishna by hearing hamotzie even if he has not yet washed. Therefore, he claims that the correct custom should be for each person to recite his own hamotzie even on Shabbos.