I don't know how the START pushbutton works, if at all. Forcing pin 1 of the op-amp to 0V will probably force its output either high or low (I don't know which), but I don't see how this would "start" the charge process. The circuit will start charging when a battery is connected and starts drawing current.
So the circuit is a current and voltage regulator that limits the charging current to about 380 mA by controlling the output voltage. The minimum output voltage is about 13.8V (13.9V minus the 76 mV dropped across R6) and the maximum output voltage is about 16.4V. This is too high, even for a 13.8V lead-acid-type battery.