You can see all the logic in the original translation: I speak English and mom does not. It is all but literal. We use "нет" instead of "don't", "doesn't", "isn't", and so on, when it's a contraction of a full phrase:
- Я ем рис, а ты нет. I eat rice, and you don't.
- У тебя есть кошка, а у меня нет. You have a cat, and I don't.
- Ты бегаешь быстро, а я нет. You run fast, and I don't (I'm not fast).
I'm not really sure it's necessary here. Can't recall a rule that requires it. My guess is that it is better to use the dash with a noun (like "мама") and omit it with pronouns (like "я"). But I'm not 100% sure.
The logic behing the "—" is that this part of the sentence is contracted, you don't say "а мама не говорит по-английски".
it is the 'does' in the second part of the English sentence.
often it means 'to be: am/is/are', but here the verb suggested is 'govor-'.
the line avoids repeating the verb 'to speak' - adjusting it to suit 3rd Person usage - and avoids using another verb like 'to do: does', as was required in English.
Russian has a neat shorthand technique. English - not! ;)