|Herbarium tour, AEM 2014
Image: L. Marsh
There are some excellent videos around which answer those questions, offering a behind-the-scenes glimpse into what goes on at two national herbaria. This video is from RBGE and this one and this are both from Kew.
What goes on in any herbarium is basically the same, whether it's a national collection or a cupboard in your bedroom. You are preserving a plant for future use and reference - whether that's to help your personal ID skills or for internationally-important research. That's the 'why' of herbaria - and the supporting data you include on your herbarium label covers where the specimen was found, when, by whom, growing with what... all the things you and your fellow botanists need to know about the plant you collected in the field!