If you look at the right side of his circuit you will see that each of the three fans has a BD140 in series with it and that the current drawn by the LM3914 is from their base circuit, so there is considerable current gain available.
Assuming each fan required 120mA for full speed and the BD140's have an Hfe of 100, then the total base current for full fan speed will be;
3 * 120/100 or 3.6mA,
three segments worth with 2* 5k = 10k between Vref and ground.
The current scaling for the LED drivers is determined by the overall value of the two pots, not their setting. Their settings determine the bottom and top of the LED range, that is the temperature where fan current starts to flow, and the temperature where it reaches its maximum value.
Iref = Vref/Rref = 1.25/10k = 0.125mA
0.125mA x10 internal scaling factor = 1.25mA/segment
I have a video here by JCM of his adaption in operation but it's 6 megs and too big to attach.