Voltage and current are linked. You can get down and dirty with the how/why etc. if you like, but I tend to just use ohms law and work out what I need... or if you're really lazy, Google an LED resistor calculator.
Leds are current controlled devices.
The outputs of the LM3915 are configured as constant current sources.
The amount of current can be set with the resistor between pin 7 and pin 8 of the LM3915 and will be about 12.5/(the resistance value between pin 7 and pin 8) in mA.
So the 1K resistor you are using will give a led current of about 12.5 mA.
If your LED is rated 20mA that is the maximum current it is designed to withstand in continuous use. It is good practice to drive components at somewhat less than maximum. 12mA will be fine.
The lower the current you put through it, the longer it will last (although you're talking years - literally). Try it at 20mA, and see if you prefer that bit of extra brightness. In practical applications, max brightness isn't always good. I quite often have to modify stuff on stage with the band to reduce the brightness of LEDs, because hey can be blinding! LOL!