Jasper S. answered 08/01/19
I teach for success in most any subject with background and experience
So, this is some colloquialism at work here and doesn't translate perfectly into the rules of English. I'm going to try my best to explain this. It's like hearing someone say "I'm tossing on the hat and getting to work". Harry is saying that he's putting on the cloak and (setting or going) off to find out what Malfoy is doing.
It's a little bit of word play on Rowling's part. She abbreviates the "off to find" from "going off to find", which might be the word you're missing.
The word "It'll" is sort of used in terms of "I will be" and "This situation is going to to be".
Basically the sentence is saying "It's time to put on the cloak and go and find out what Malfoy is doing". It's just more fun saying "On with the cloak, off to find out".
I hope that helps.