If something purportedly happened, it may or may not have actually occurred. If your brother purportedly knows nothing about the cookies you left on the kitchen counter, it means he claims to be innocent of cookie theft. Whether or not you believe him, well, that's another story...
When you use purportedly it means you only know what you've been told, or what appears to be the case, not necessarily what the entire truth is. A dog that purportedly doesn't bite is still scary when it's barking at you. And although your friend is purportedly only joking when she says she hates your haircut, it may still hurt your feelings. You can also use the synonyms "supposedly" or "allegedly."