A leader who hires his own personal clown will likely double his entertainment. Clowns are funny and make people laugh, so the leader decided to hire one. Since that time, the leader’s amount of mirth has doubled.
This also means times, which makes sense when you think about the kind of math you'll need to double something. What is A * 2? It's A doubled, first of all, but you're using multiplication. You're "timesing" something.
When the clown stops being funny, it’s time to say bye (ばい). At first, the clown doubled the leader’s entertainment, but now the clown is just annoying. It’s time to say bye-bye.