Why was a transistor used in this circuit for setting the backlight of an LCD display?
Without information what is behind the connector it is hard to say, but probably the transistor is used to pull a pin low or to allow a current to flow (probably for the backlight).

Probably the backlight is directly connected to power on LCD itself. To allow current to flow (the backlight to light) you need a direct path. You can establish this path if the transistor is conducting. If it is blocking the path, no current will flow and the backlight will be dark.

The values itself are not important here. The Pin LCD Backlight is probably a 5V or 3V3 pin, this yields approximately 1mA into the base of the transistor through R1 if the pin is high, causing the transistor to conduct. If the pin is low, the transistor will not conduct. R2 is a pull-down, in case the Pin is not defined (e.g. during start-up). It will make sure the transistor doesn't conduct even if the pin is in high impedance state (this is the default for most GPIO Pins on most microcontrollers).