Almost, The diode is flipped/reversed. (I think you would have quickly figured it out)
The diode is used to isolate D10 from the circuit when D10 is high.
The pullup resistor is used to turn on the transistor.
The backlight will be on if D10 is input or HIGH.
Setting D10 to low turns off the backlight.
The diode allows D10 to yank the pullup down to gnd to turn off the transistor when D10 is low.
It works but really isn't a good design, I wouldn't design a circuit like this.
I did it this way because I couldn't easily desolder the LCD on the shield I had to replace
the transistor with an FET. (The transistor was underneath the soldered LCD module)
--- bill