Delay between toilet flushing and filling
That valve is necessary for whenever you need to service the plumbing inside of the toilet, it turns the water supply off. In the cistern on the other end of that pipe coming from the valve, there is a "fill valve" mechanism of some sort, tied to a float. When the water drops down into the bowl to flush, the float drops and enables the fill valve to refill the cistern, until the level raises again and it shuts off.

Most likely your fill valve is getting old and is sticking when the level drops, so it shows up as a delay in that the float is not falling when it should. This would be an indicator that it is close to failing completely and needs to be fixed / replaced.