Both wires move

Neither wire is ground

In most circuits one wire of a pair is ground and the other carries the signal. Not here. The NS4168's output is bridge-tied (BTL): AUDIO + and AUDIO - both switch, and the speaker hangs between them.

Each wire switches between ground and the supply at about 430 kHz, far above hearing, and the width of the pulses follows the sound. The two move in opposite senses, and the speaker feels only the difference between them. Its own coil is an inductor, which smooths the fast switching into the slow movement of the cone. That is why the chip is called a filterless Class-D amplifier.

The board adds a ferrite bead and a 2.2 nF capacitor on each wire to keep the switching from radiating. They are there for interference, not for the sound.

What never goes on them

• GND. Either wire to ground is a short every time that wire goes high. The chip protects itself and shuts the output off, so you get silence until the wire comes away. That includes a speaker lead that brushes the ESP32's GND.
• Headphones. This is a speaker-level switching output with no ground wire. Headphones belong on the TK103.
• Another amplifier's input. The input expects a small, smooth signal against ground. This is neither. Feed an amplifier from the TK103 instead.

The one thing that goes on AUDIO + and AUDIO - is a bare speaker, 4 or 8 Ω, on its own two wires.