Purpose of capacitors in this project
A capacitor is a device that can store (and release) charge. It is made up of 2 metallic electrodes (or plates) that are separated by a thin dielectric material.


  When you apply a voltage over the two plates, an electric field is created. Positive charge will collect on one plate and negative charge on the other. - build-electronic-circuits.com


In short, a capacitor lets the AC component through and blocks the DC component.



Capacitors can be used in the following ways:

Decoupling Capacitors

(Also called Bypass capacitors)  


  A decoupling capacitor’s job is to supress high-frequency noise in power supply signals. They take tiny voltage ripples, which could otherwise be harmful to delicate ICs, out of the voltage supply.  




Image reference http://www.seattlerobotics.org/encoder/jun97/basics.html


  In a way, decoupling capacitors act as a very small, local power supply for ICs (almost like an uninterupptable power supply is to computers). If the power supply very temporarily drops its voltage (which is actually pretty common, especially when the circuit it’s powering is constantly switching its load requirements), a decoupling capacitor can briefly supply power at the correct voltage. This is why these capacitors are also called bypass caps; they can temporarily act as a power source, bypassing the power supply.  
  
  Decoupling capacitors connect between the power source (5V, 3.3V, etc.) and ground. It’s not uncommon to use two or more different-valued, even different types of capacitors to bypass the power supply, because some capacitor values will be better than others at filtering out certain frequencies of noise.
  - sparkfun.com


Power Supply Filtering

DC power supplies can be quite noisy on their own. Take the output of a bridge rectifier for example, there is a lot of ripple on the output. Capacitors are a perfect solution for reducing the ripple/noise of power supplies.



Image Reference: http://www.electronics-tutorials.ws/diode/diode_6.html


  Capacitors are stubborn components, they’ll always try to resist sudden changes in voltage. The filter capacitor will charge up as the rectified voltage increases. When the rectified voltage coming into the cap starts its rapid decline, the capacitor will access its bank of stored energy, and it’ll discharge very slowly, supplying energy to the load. The capacitor shouldn’t fully discharge before the input rectified signal starts to increase again, recharging the cap. This dance plays out many times a second, over-and-over as long as the power supply is in use.


Also see Keelan's answer over at electronics stack exchange.

Signal Filtering


  You can also combine capacitors and resistors to form filters that target specific frequencies. - build-electronic-circuits.com
  
  They can block out low-frequency or DC signal-components while allowing higher frequencies to pass right through. They’re like a bouncer at a very exclusive club for high frequencies only. - sparkfun.com
  
  For example in an audio system you can target the high frequencies to remove them (e.g. in a sub-woofer). This is called a low-pass filter.  - build-electronic-circuits.com  




Image Reference: Wikipedia



Back to your application. 

Motors are notoriously noisy. Your mentor has recommended a capacitor to decouple the noise of the motors and protect the IC from voltage drops/spikes.

Side note: D1 in your schematic is a power indication LED. It needs a current limiting resistor in series with the LED to limit the current and prevent the LED from burning out.



Resources:  


https://learn.sparkfun.com/tutorials/capacitors
https://www.build-electronic-circuits.com/how-does-a-capacitor-work/
Basic Circuits - Bypass Capacitors
wikipedia.org/wiki/Capacitor
wikipedia.org/wiki/Decoupling_capacitor
Youtube - How a Capacitor Works - by Dr. Oliver Winn
explainthatstuff.com/capacitors