In the sentence at top, the speaker says "this internship". That means that he is talking about one specific internship. In this example, the speaker was just telling his friend about the internship that he wanted to get. So his friend knew exactly which one he was talking about.
If the speaker wasn't talking about one specific internship, but about any one, he would say:
It's super important that I get an internship.
And if the speaker wanted to talk about one specific internship, but his friend didn't know about it, he would say:
It's super important that I get this one internship that I'm applying for.