“You can't just go around using people's stuff without their permission!”

You're in a band. You've been looking for your guitar, and are surprised to discover that your bandmate is playing it. You're angry that he took the guitar without asking you first. You say:

You can't just go around using people's stuff without their permission!

You can't just (do something)

Use this expression to scold someone who's doing something bad but seems to think that their actions are totally normal:

You can't just call in sick whenever you feel like it.

You can't just walk in without an appointment.

go around (doing something)

The phrase "go around ___ing" describes someone doing something bad again and again, with different people.

For example, if someone has been spreading false rumors about you, you can say:

She's been going around telling everyone that I slept with Kyle. That's a lie!

If your husband or wife has been spending a lot of money on useless things, you can say:

You think you can just go around spending money left and right? We've got bills to pay, damn it!

It is possible to use "go around ___ing" to talk about someone doing a good thing with many different people, but it's rare. Even if you're talking about the person doing something good, it sounds like you're complaining:

He's always going around telling everyone what a great job they're doing.

people's stuff

In the example, the speaker says "people's stuff" instead of "my stuff". This makes it sound like a general rule. General rules are often expressed in terms of "people":

If you want people to like you, you have to be nice to them.

use (something) without (someone's) permission

When you ask someone if you can do something (like "Can I use your guitar?"), and they say "yes", you have their "permission". 

Using something "without permission" means that you didn't ask the correct people before using it.