The transistor tester is a discrete circuit which is only testing the functioning of a bipolar junction transistor. If the corresponding LED is blinking, the transistor is working. If both are blinking or none, then the transistor is dead.
The circuit revolves around the CMOS 4093 quad-NAND with Schmitt-Trigger inputs. One gate is generating a square wave with the help of an RC-filter. A following gate is just inverting the signal for better signal integrity and less distortions. One gate is unused and the other is inverting the (a bit) cleaner signal again. If no transistor is connected, the inverters are activating the LEDs in an alternating fashion. If a NPN transistor is connected, he shorts out one part of the square wave and thus only allows the NPN LED to blink. A PNP transistor is doing the same, but with the other LED.
This only works if you press the button, to save power.
This is an example of a simple Full-NAND logic circuit. This way you don’t need multiple different kinds of gates and ICs.
And here is the schematic
And a few pictures