This is a item which used to be in every household, a pottery chopstick drainer, a bit like the cutlery drainer which is now replaced by the plastic draining tray. I must admit I am part of the plaster tray generation and have never seen the chopstick drainer in action.
On the drainer is written 百子千孫, hundred of sons and thousands of grandsons, a blessing of fertility. At a time when mortality rate is high, a large family to continuation the bloodline is the most important aim in life, though it does such a priority in life as much (particularly with China’s one child policy) its still seen as a welcome blessing for the family.