"Der Mann trägt ihre Kleider."
Translation:The man wears her dresses.
Well neither would be said in real life without context. Both 'her' and 'their' require some form of identification, for example "Do you see that lady? The man is wearing her dresses", which would be "Siehst du die Frau? Der Mann tragt ihre Kleider". In this case it would be very clear that it is 'her' and not 'their'.
EDIT: Whoops, sorry, I got Ihr (you) and Sie (they/you formal) confused. This language's pronoun system drives me quietly insane ;-) You may ignore the rest of my comment.
As far as I can tell, ihr should only be capitalised when it's formal, where as addressing multiple people in normal context would use just plain old ihr with a lower case "i". So it's more complicated than that (which is probably why Duolingo hasn't yet touched on this topic).
Frustrating...when the translation underneath the words clearly says that "kleider" can mean clothes! And why is it that "trägt" cannot mean carries? I thought the translation would be a normal nowadays situation from a mall,where a man carries him woman's clothes and bags :)))))))