Does any one know what these branches are and what their purpose is (like herbal)? Is it in Japanese culture or any other culture?
NSFW (Nudity) content and manga spoilers ahead.
Those are White Birch Branches. It's also mentioned in the manga during Chapter 20.
I'm not sure of their exact use but according to the Wikipedia page of birches,
fragrant twigs of silver birch are used in saunas to relax the muscles.
This seems to coincide with what is shown in the manga/anime, as it looks like it is used to relieve exhaustion and provide relaxation.
The term for these is a bath broom.
Or "banny venik" (from банный веник) in Russia.
Or "vasta" or "vihta" (depending on the region) in Finland.
They're mostly used in heated bath houses and, prior to refridgeration, were a rather seasonal luxury. You use them, quite simply, just to hit yourself with in a hot bath house.
Despite how weird (and painful) that sounds, it's actually a rather pleasant feeling experience, mostly compared to massaging yourself, and provided that the branches are fresh enough, you can use a fair amout of force without hurting yourself. The leaves, that should not be dry (or you're doing something horribly wrong) cushion the impact nicely. They're also often use wet.
Here's also a short video about the actual use of the brooms (1:55 for the actual "application"): https://www.youtube.com/watch?v=w4ZILOglcHE
Besides all that, the fresh birch also leaves behind a rather pleasant scent.