if you consider a moment before the sample switch closes and a positive input of 100 micro volts, with no diodes. The switch is open so the first op-amp is running without any feed back so its output stage switches to Vcc and saturates and charges up any parasitic capacitance to + Vcc. When the switch closes this charge then charges up the storage cap, and the output stages take some finite time to come out of saturation. Adding the diodes, stops the output stages saturating because as soon as the output exceeds +- .7 of a volt, the gain of the opamp drops to unity. This means that the output of this amp is curtailed to +- .7V, which also reduces the charge on the parasitic capacitance. This output then goes to +- 0V after some time when the second opamp output follows the storage voltage.
" the range -2 to 2V. Is " no its -2 -> +5V
Frank