That message is just a general warning and should be the moment where you should think about “what will happen next” and that highly depends on the report that you read. Let me give you a few examples:
A report about crashing Internet Explorer 11 by visiting a specially crafted html page
Be prepared that if you open the test URL in IE 11 the IE will crash and close unexpectedly.
So if your day to day browser is e.g. Firefox and you use IE just to open that single website you can simply try that out. If you use IE as your main browser (hopefully not) it would be better to e.g. create a new windows user and try this out in a separate user account to prevent your browser profile from getting corrupted.
Side-note: In this case you can’t even try it under linux because IE is windows only.
A report about an XSS in Firefox below version 50
A normal XSS demonstration will only show something like an alert and therefore there is nothing to worry about and you can simply try it out with your main browser as long as you are using a version smaller than Firefox 50. If you already have a newer version of Firefox your main problem will be: how do I get that older version running on my computer… maybe you can use a portable version or you spin up a virtual machine where you install that specific version.
A report about a system crash for Chrome up to version 15
A crashing system could lead to all kind of problems like corrupting open files, loosing unsaved data etc. so the easiest thing to try this out is a virtual machine or a dedicated (old) lab computer where you don’t run any important stuff and can mess around freely without risking the loss of important data.
You can get away without a virtual machine and don’t need linux but i would highly recommend to try out virtualisation software like Virtual Box because they make your life so much easier. You can for example create a snapshots of your virtual machine… so before you install a crazy old version of some software to try something out, you simply save the state of that virtual machine, start to fiddle around with that old software and as soon as you’ve finished your research, you can simply restore that saved snapshot. After that your virtual computer is in a clean state again, ready for next tests.