I'm kind of curious as to why there is a 0.1uF capacitor on this Arduino Pro Mini (https://www.arduino.cc/en/uploads/Main/Arduino-Pro-Mini-schematic.pdf). It seems like it would work just fine without it.
Most USB to serial chips set the DTR to LOW on connecting, but it keeps it LOW.
This would reset the arduino, but since the DTR never goes HIGH the Arduino will never get out of this reset state, and not run your code.
The capacitor will AC couple the signal. The the DTR goes low, the other side of the capacitor will go LOW too, but the pull-up resistor will slowly charge the capacitor, till the reset pin is HIGH again.
So it will convert a LOW transition to a LOW pulse.