Help me understand this

I have a P-PKG1W-WH and today it stopped working. I'm no electrician, but I figure out what was wrong after a long process of elimination.
Help me understand why it didn't work.

The switch controls a trio of LED lights in a ceiling fan. The fan side is controlled by a different switch.
Today I noticed the 6 small led lights on the switch were off. When I tried to turn on the light it wouldn't work.
Come to find out my wife turned off the light by using the string on the fan.

Help me understand why turning off the light by the string cuts off power completely to the switch.

Thank you

The Caseta dimmer requires a lighting load in order to create enough current to power the microchip in the dimmer, which in turn powers those 7 indicator LEDs on the dimmer. If you remove that load, then you've broken the circuit loop and there is no current running through the dimmer and no way to power it. This is why those LED indicators were off.

Yep, remove the string/chain from the fan (leaving it in the on position first!). Otherwise you cut off all electric to the lights, and the switch won't be able to control them. With some fans it's possible to retract the short length of chain or cord into the housings. That's often worth doing to head-off any future issues.