When the panel isn't generating, the entire circuit is off and there is absolutely no current drain from the battery. When the sun gets up and panel starts producing at least 10 Volt, the LED lights and the two small transistors switch on. This powers the regulator circuit. As long as the battery voltage stays below 14V, the operational amplifier (which is a very low power device) will keep the MOSFET off, so nothing special will happen and the panel current will go through the Schottky diode to the battery.
When the battery reaches the trigger voltage, which is nominally 14.0V, U1 switches on the MOSFET. This shorts out the solar panel (a condition that is perfectly safe), the battery no longer gets charging current, the LED goes off, the two small transistors go off, and C2 powers the regulator circuit while slowly discharging. After roughly 3 seconds, C2 has discharged enough to overcome the hysteresis of U1, which switches the MOSFET off again. Now the circuit will again charge the battery, until it again reaches the trigger voltage. In this way, the regulator works in cycles, with each OFF period being 3 seconds, and each ON period lasting for as long as necessary for the battery to reach 14.0V. The pulse length will vary according to the current demand of the battery and any load connected to it.
Building this circuit is very simple. All components are widely available, and most can be easily replaced by other types if necessary. I would not advice to replace the TLC271 nor the LM385-2.5 by different ones, unless you know very well what you are doing. Both of them are low power devices, and their power consumption directly defines the OFF time of the regulator. If you use replacements that have a different power consumption, you will need to change the value of C2, adjust the biasing of Q3, and maybe even then you might run into unexpected trouble.
The MOSFET can easily be replaced by any type you like, as long as its...