It's a comfort mechanism.
Although you see a calm collected presenter, what is going on is that they are hearing what is called 'gallery sound' through an ear-piece. Because live TV is so fast moving, there is no time to swtich individual ear-pieces on and off, so everyone on the studio floor hears everyone else's instructions, as well as their own.
That means that the presenter is hearing instructions to maybe a dozen people, cameramen, floor-walkers, other presenters, etc., as well as instructions to them personally. So, the presenter has to learn not only to 'tune out' instructions which are not for them, but to make sure they hear and respond when an instruction is actually for them. They do this while simultaniously talking to camera, reading an autocue, and timing links to fit in with filmed inserts - oh, and appearing totally natural of course!
This takes massive concentration, and in order to stop subconcious hand movements, tapping fingers, playing with rings etc., presenters hold onto a biro, or a paper, which keeps their hands still.
It may be an irritation, but that is the reason for it.