Mourning means grieving. "Mourning has broken" means grieving has stopped. "Mourning" is a homophone of "morning". "Morning has broken" means the day has just started. Morning is a time of day when the sun rises. Easter Sunday is recognized by Christians as the day Jesus rose from the dead. This joke is playing on the phrase "morning has broken" by replacing "morning" with "mourning". 
Mourning has broken on Easter Sunday .