This crystal tester is very straightforward. Fitting a crystal or switching on the supply voltage generates a ‘start pulse’ resulting from the fact that the crystal briefly pulls the voltage on the base of T1 low. This directly affects the operating point of the transistor via feedback capacitor C1, with the result that the transistor starts oscillating.
Resistor R2 limits the maximum operating current of the transistor. A 100-pF capacitor (C2) is connected in parallel with R2 for decoupling, and capacitor C3 is used to prevent the DC voltage on the emitter from appearing at the output.
An AC signal will, therefore, be present at the output if the crystal is OK. You can put together your own indicator circuit to make this visible, such as an HF probe connected to a meter or a transistor with an LED.
Another tip: if you connect two LEDs in reverse parallel in series with the ground lead of the crystal, they will both light up when the crystal oscillates.