How does the range of bluetooth increases by adding an antenna?
Generally spoken, replacing an antenna changes antenna characteristics. The efficiency of the antenna which is on board can only be improved if it is a bad design. This might be if, e.g. the antenna design was subject restrictions like space on the PCB or price of the PCB-base material inflicting a bad \$\epsilon_r\$ or somthing.

Antenna characteristics can be changed concerning directivity. This can increase range in certain directions but at the cost of range in other directions.

If the matching of the antenna is modified, power consumption can be increased as well as decreased. Normally the built in antenna should be matched quite good, so the radiated power will probably decrease, when you change it. There might be increased losses, but it's also possible that the overall power consumption decreases. 

It is not easy to predict the changes in behaviour. Because changing the directional characteristics can increase EIRP and sensitivity very strongly. 12 dB or more is possible at the cost written above. 

Length of wire inflicts the matching most. But there is no linear connection between length and transmission lines, because antennas only want to resonate at multiple frequencies of their base frequencies (simply spoken) which is connected to the length of the antenna.