This is a pronom tonique. It does serve to emphasize or create contrast, and it does go outside the verb phrase.
The subject and object pronouns in French do not take emphasis well. They’re considered a kind of clitic — a verbal appendage that happens to be separate from it rather than their own word. So you have to add something else if you want to put stress on it.
I see that you have experience with Spanish. It’s analogous:
Me gustan las manzanas. J’aime les pommes.
¡A mi me gustan las manzanas! J’aime les pommes, moi !
Whereas you wouldn’t add emphasis by stressing the original object pronoun:
Te gustan las manzanas. Tu aimes les pommes.
A ti te gustan las manzanas. Toi, tu aimes les pommes.
One real-world example I like to cite is John 3:30, in which John the Baptist contrasts his trajectory and Jesus’. In one French translation, he creates that contrast this way:
Il faut qu’il grandisse et que moi, je diminue. (Segond 21)
In the example you quoted, the emphasis is somewhat subtle and hard to translate. It would be a bit like saying, "And how did you think I would react?" or "You didn’t consider what that would do to me." (And they say French is less concise than English!)