Someone who behaves like a stuffed shirt is rather conservative and might even have reactionary attitude. At any rate ‘a stuffed shirt’ is a boring person I’d say.
Here is the sentence you are referring to:
It’s very difficult to make him enjoy himself, relax and take it easy because he always behaves like a stuffed shirt.[YSaerTTEW443543]
Just to add to what Torsten has written - a stuffed shirt is someone who always wants to do the correct thing and can never really take it easy because he/she cannot relax. The expression comes from the dummy or figure you see in a clothes shop window showing the clothes which are for sale. Obviously this figure is stiff and and doesn’t move!