One of the uses for a unity gain amplifier (voltage follower or buffer circuit) As mentioned in the documentation section, is to drive circuits without loading them, RC discharging to get a time delay is such a circuit, if say for example an RC discharge (decay) circuit was built with a 10 second delay, then if a large led or maybe some other circuit was attached without any sort of buffer it would just load the RC network and pull the current straight out and the time delay would not work correctly. (try it without this buffer) it lasts around just short of a second. Using a voltage follower a buffer circuit say an op-amp, as it has very high input impedance or resistance, would not pull hardly any current from this circuit, thus allowing the decay of the capacitor and time constants to work, this would decay exponentially over 10s and the voltage follower, would just follow it. The time constant for resistor and capacitor in series is T = R.C where R is in ohms and C is in farads, and T is in seconds therefore R(100,000 x C(0.0001) = 10 seconds, so a 100k resistor and a 100uf capacitor would keep an led bright for 10 seconds then after this 1st time constant should loose the brightness as the voltage has reduced by 63 percent, and from a 5 volt supply this is scarcely enough to keep an led lit, particularly if it is green as their forward voltage drop is 3v-3.5v, for a red led been typically 2v-2.5v this keeps on slightly longer. After five time constants 5t the capacitor is said to be fully discharged. So exponential decay is apparent here as the voltage drops off quick after the first time constant (1t) and slows through the rest of the time constants with respect to time.
The diode on the output of the op-amp is to act as a voltage dropper, due to offset voltages, (for this circuit anyway) just to make sure the led does go out after the required time, and not just look dim. Diodes can be used as voltage droppers in series, they drop 0.6v-0.7v, or or 0.3v for schottky diodes or a little bit higher depending on current draw through them. So for five normal rectifier types these could cause a voltage drop of 3v-3.5v for low current circuits. And maybe up to 5v for larger current circuits.