The issue stems from the way that the Apple Watch senses your heartbeat. According to Apple, the back of the Watch rapidly flashes green and infrared light at your skin, which gets absorbed or reflected by your red blood.
When your heart beats, there is more blood in your wrist, and there is less blood between beats. By sensing the timing between your heartbeats, the Apple Watch can calculate your heart rate.
As it turns out, solid-colored tattoos -- particularly red ones -- also absorb the green light and reflect red light. Black tattoos, which absorb both green and red light, can also screw up the Apple Watch's heart rate sensor.