When a device gets jailbroken, it simply means that someone modified it so that they had unrestricted access to new features and apps the normal version of the device wouldn't allow. Jailbreaking a Fire Stick is a little different.
A jailbroken Fire Stick already has a media player software downloaded onto the device. According to Grounded Reason, people who have jailbroken Fire Sticks usually have a software called KODI, which is free and anyone can download. It is legal, but this is where the Fire Stick starts to get messy.
A Fire Stick with KODI can essentially stream content from anywhere on the Internet. So, if you're trying to watch The Notebook for the millionth time, KODI can find a free online version from an illegal website. You can also download an app to watch Netflix for free, which sounds ideal, but is also illegal.
Jailbroken Fire Sticks have become slightly trendy, so some people now charge anyone who wants to jailbreak their device. This is totally a scam, since anyone can easily do it themselves for free. Grounded Reason lists a step-by-step guide for how to do this, so don't let anyone trick you into paying more than the device itself to simply download KODI.
Moral of the story: Fire Sticks are cool and affordable, but you need to be wary of how you use the device if you decide to get one. If you want to go the jailbreaking route, make sure you don't cross the line into illegaility. And if you're not sure whether something you're watching on your Fire Stick is legal or not, you should probably opt for something else.