Like kinjalgp said , the transistor circuit is used as a constant current source for charging the capacitor .The opamp in the bottom schematic is
called a voltage follower which as a gain of 1 so there isnt any amplification, its purpose is to isolate the output from any connected load.
The constant current source is what gives the nice straight lines in the ramping voltage seen in the posted drawing. If the 555 was connected in its regular astable or monostable configuration, the ramping voltage would be more of a smooth exponential curve.