Why does my Esplora only send serial data when the serial monitor is on?
Esplora is based on a MCU with native USB. The native USB reports connected only when it really connects. You have in setup() a while loop until the USB Serial is valid. The skech waits there for connection. Comment out the while (Serial) loop.

The TX and RX LEDs of Esplora are connected to the native USB port Serial interface. They will not light for the Serial1 interface.

You should disconnect the Uno from computer to not disturb the Serial interface with USB connection. Or use SoftwareSerial to communicate with Esplora and have debug prints to Serial Monitor.