Why does this transistor circuit light the LED?
Shouldn't it only light if I apply + voltage to the base?
No. If the base is disconnected (or the 1K resistor attached to the base is disconnected), is it still possible that the led lights up, even if there's no base current, due to leakage current in the transistor die, or due to the digital output from the eps8266-12 not being exactly at 0V.
It would be adviseable to connect a resistor (about 10k) from the base to 0V. This resistor provides a path for the leakage carriers to escape to ground, and this will stop the transistor conduction when there's no voltage applied to the 1K resistor.
Also, you're connecting the led incorrectly. The led connects the 12V directly to the transistor's collector, this is wrong. If the transistor conducts (if there is base current), the led will fry. You should put a load resistor (about 470 ohms) in series with the led. This is my suggestion of improvement to your circuit:



simulate this circuit &ndash; Schematic created using CircuitLab
Thanks to the user tlfong01 for pointing an error in the words that I used to describe the base-resistor.
Since English is not my home language, I ask for any suggestions to improve the text and thank you in advance.