The wording "hamotzi lechem min ha-aretz" is debated in the Talmud. Rabbi Nechemiah preferred motzi ("brings forth"); the rabbis preferred hamotzi (with the definite article). The halacha follows the rabbis. The phrase echoes Psalm 104:14, "motzi lechem min ha-aretz", which is itself a paraphrase of the agricultural cycle described in the Psalm.
In Talmudic-era life, bread defined a meal. Other foods could be eaten as snacks, but if bread was set on the table, the gathering was a formal meal. The rabbis encoded this in halacha: bread requires Netilat Yadayim, HaMotzi, and a full Birkat HaMazon after, none of which are required for other foods. The bread blessing also "subsumes" the other foods in the meal: you don't bless the soup separately, since the soup is part of the bread-meal.
This is why Shabbat tables have two challot, they're the formal anchor of the meal, and the double bread also recalls the double portion of manna that fell on Fridays in the wilderness (so the people wouldn't have to gather on Shabbat).
1. Some sprinkle salt on the bread before HaMotzi (or after), echoing the Temple offerings, which always required salt. The custom also says salt averts misfortune, "the salt of the covenant" (Lev 2:13).
2. After HaMotzi, take a piece for yourself first, then pass to others. Don't pass the bread before tasting it.
3. The bread should be whole (a complete loaf, two on Shabbat) when blessed, then cut after. Some communities cut a small notch in the bread before saying the blessing so the cut motion is already started.