Actually, although our RoboGirl can sound quite peculiar sometimes, in this particular sentence the words are perfectly clear. It may be that your ear is not yet attuned, which is a situation that will improve with practice.
I think it is the playback function, not the recording, that is garbled, because when I hit replay the part I couldn't hear the first time sounds fine, but a different part of the sentence becomes garbled.
Or it may be that not every device has the same precision, or the same version of the app or OS. Actually, this may not be the fault of the learner's ear.
"Mon peuple" refers to only one (group of) people....my people. "People" is a word which refers to many individuals but it is spoken of as a single thing. I would say it behaves the same way as "le raisin" refers to "grapeS" and not "a grape". One could say "the peoples of the world" if referring to the many different "people" groups which comprise the entire population. But in this sentence, that is not the case. So, "You are my people" becomes "Vous êtes mon peuple", not "...mes peuples."
Basically, a group of people is always treated as a singular entity and masculine, unless everyone in said group is a woman. Even if there is only one man in the group, it also remains masculine.
I think what's significant here is that the use of "un peuple" as a collective noun is more common than the parallel use of "a people" is in English.
Probably this is because of the fact that "people" is also the usual plural form of "person". You can say, for example, "The French are a proud people", or "The French are proud people"; there is a slight difference in meaning, but not much.
In a sentence like the one we have here, "You are my people", the word "people" could technically be either the collective or the plural, but I think you'd find that people generally hear it as the plural.
I base that assertion on the observable fact that in general, Anglophones want to translate it as "mes peuples", or "mes peuple", not "mon peuple".
"Folk" without "s" is similar to "people" as a collective singular. It's archaic. Nowadays you see it mainly in expressions like "folk music". Maybe one reason we don't use it much any more is because it's too much like the German ein Volk, ein Fuhrer?