What does it mean to show off when the tiger is away and I am a sick cat?

This sentence means not to underestimate a person.

"When I am a sick cat, the tiger doesn't show arrogance" is slang in China, which means not to underestimate or belittle someone's strength or ability, even if it seems gentle or harmless on the surface. This sentence is often used to warn or remind others not to judge a person's true strength only by appearance or initial impression. Just like a tiger, even if it looks quiet and docile at ordinary times, once it shows its strength, it will show amazing strength and majesty.