The next time you repeat this (and in general, you do everything right to re-enforce new vocabulary) I will try to indulge your rigidity because my lesson is taking more time than I like and I realize you will repeat this until doomsday—but I would certainly not do so "in real life".
I'm interested to know what country the speaker is from. None of the sentences sound like any of the Arabic I've ever heard. The most i've heard is Saudi, Lebanese, Iraqi and Palestinian/Jordanian. Also some Syrian, Omani and Dhubain (? if you're from Dhubai are you Dhubain? I don't know!) And a tiny bit from Morocco. I think maybe I haven't heard Egyptian- is the speaker from there?!?
The speaker here is a machine actually (and hence lot of mistakes happen in the audio grammar- or pronunciation-wise). However, from the tone, I can tell that the original speaker or the base is Egyptian (I mean the sound or the voice of the speaker and not the dialect - the sentence here sounds proper in Arabic and not Egyptian dialect).
because عليكم is the preposition على (3alá) which means ON/UPON
the suffix ـكم (-k u m), which stands for (you/plural).
Combining the two it becomes (upon you).
The suffix here is plural but it is commonly used as it is without distinguishing a single person or a group of people usually, and even regardless of the gender of the people you are greeting.
hmm to what I know "peace be with you" is used as a farewell gesture in English. On the other hand السلام عليكم can be used for both but it is mostly linked to greetings more than farewell (i.e. saying goodbye). This is one side of the story.
On the other hand, in the English sentence there is "with," while the Arabic version has "on/upon" على (attached to the pronoun suffix ـكم = عليكم).
In Arabic, the typical and mostly used farewell is مع السلامة (by both, the leaving person and the person staying in place), and that translates literally as "with peace/safety".