Amcha is a compound word: Am is "people" and cha is "your". Thus: your people.
It is probably taken >from the Book of Exodus, following Israel's sin of
worshipping the golden calf. God says to Moses: "Look what your people have done!?
Moses tries to calm God down, saying: "Remember, that this is Your People!?
(Exodus 32: 7-11).
Amcha, in later Jewish use, was not understood literally ("your people") but was
used, informally to be "our People". But "amcha" also came to have a quality
associated with it, based on another usage involving the word, "Am". "Am
ha'aretz", literally, "people of the land", that is, farmers, in later usage
meant an individual who was ignorant. Amcha comes to mean "the common folk".
When one writes a letter for amcha, then, he's not just writing for "our
people", one of us, but writing for the simple folk among us, who aren't