Battery Recharge Circuit Help
This is not a great circuit in that there should be a 1k or so resistor in series with the base of the BC547 transistor.  But that said:

Notice that the emitter of the transistor (NPN) is connected to the most negative point in the circuit: the (-) input from the DC power supply.  Also note that the base (and current-sense resistor) is connected to the battery (-) terminal.

As the battery draws current through the circuit, a voltage proportional to that current is developed across the resistor at the transistor.  When that voltage reaches about 0.65 Vdc, the transistor begins to conduct and pulls the Adjust lead of the regulator towards ground.  

The regulator is designed such that the output terminal is always about 1.2V above the Adj pin.  When you pull the Adj pin towards ground, the output voltage drops.

The circuit is a classic negative-feedback regulator.  The transistor will adjust the regulator voltage to keep the output current right at the point where the transistor is conducting.  As mentioned earlier, that corresponds to about 0.65 Vdc across the current sense resistor.

The reason I mention there should be a 1k or so resistor in series with the base of the transistor is that if the output leads are shorted, the circuit can't respond quick enough to drop the output voltage.  This results in a substantial current into the base of the transistor - this usually destroys the transistor.

Been there, done that.