“I don't want to get into it right now.”
You're upset because of a problem that you're having with your visa paperwork. You're talking to a friend and she can tell that you're upset. She asks what's wrong. You say this because it would take too long to explain and you're too upset to talk about it.
I don't want to get into it right now.
To "get into" a topic or a story means to talk about it in detail. For example:
Jordan and I got into a long debate about whether to buy a Mac or a PC.
I'll get into that a little later.
The phrase "right now" sounds just a little friendlier and more conversational than just saying "now".