Why is the output signal greater than the input signal in this voltage follower circuit?
The AD8606 is intended for a supply range from 2.7 volts to 5.5 volts and has a maximum supply voltage of 6 volts. According to your circuit, you are feeding it with +/- 3.5 volt supplies and this means the power supply is 7 volts: -



I'm not surprised that strange things may occur with such disregard for the data sheet.

You also indicate a 0.1 uF input capacitor yet, your breadboard shows an electrolytic AND, it appears that R1 is in series with the input signal instead of acting as a light pull-down: -



So, no bias on pin 3 and busting the absolute max ratings doesn't bode well for testing the AD8606.

I also don't see any supply decoupling capacitors. Typically 100 nF from each supply to GND are required for stable operation.