The diode is to provide a safe path for the inductive kickback of the motor. If you try to switch off the current in an inductor suddenly, it will make whatever voltage is necessary to keep the current flowing in the short term. Put another way, the current thru an inductor can never change instantaneously. There will always be some finite slope.
The motor is partially an inductor. If the transistor shuts off quickly, then the current that must still flow thru the inductor for a little while will flow thru the diode and cause no harm. Without the diode, the voltage across the motor would get as large as necessary to keep the current flowing, which would probably require frying the transistor.
A small capacitor across the motor will reduce the speed of the possibly fast voltage transitions, which causes less radiation and limits the dV/dt the transistor is subjected to. 100 nF is excessive for this, and will prevent efficient operation at all but low PWM frequencies. I'd use 100 pF or so, perhaps to up 1 nF.
The resistor is to limit current the digital output must source and the transistor base must handle. The transistor B-E looks like a diode to the external circuit. The voltage will therefore be limited to 750 mV or so. Holding a digital output at 750 mV when it is trying to drive to 5 V or 3.3 V is out of spec. It could damage the digital output. Or, if the digital output can source a lot of current, then it could damage the transistor.
1 kΩ is again a questionable value. Even with a 5 V digital output, that will put only 4.3 mA or so thru the base: the voltage drop at the B-E junction ("diode") is 0.7 V, leaving the 4.3 V at the resistor. You don't show specs for the transistor, so let's figure it has a minimum guaranteed gain of 50. That means you can only count on the transistor supporting 4.3 mA x 50 = 215 mA of motor current. That sounds low, especially for startup, unless this is a very small motor. I would look at what the digital output can safely source and adjust R1 to draw most of that.
Another issue is that the 1N4004 diode is inappropriate here, especially since you will be turning the motor on and off rapidly, as implied by "PWM". This diode is a power rectifier intended for normal power line frequencies like 50-60 Hz. It has very slow recovery. Use a Schottky diode instead. Any generic 1 A 30 V Schottky diode will do fine and be better than a 1N4004.
I can see how this circuit can appear to work, but it clearly wasn't designed by someone that really knew what they were doing. In general, if you see an Arduino in a circuit you find on the 'net someplace, especially a simple one, assume it was posted because the author considers it a great accomplishment. Those that know what they are doing and draw out a circuit like this in a minute don't consider it worth writing up a web page on. That leaves those that took two weeks to get the motor to spin without the transistor blowing up and they're not really sure what everything does to write these web pages.
It looks like a common emitter amplifier using a PNP transistor so it has a negative power rail with respect to 0V. However there is a positive voltage biasing the base and this will reverse bias the base-emitter juntion turning the transistor off to a certain degree.
When an input signal is of sufficient peak-to-peak amplitude, part of the negative section of the input waveform will cause the base-emitter junction to be forward biased and the output will amplify this part of the input waveform and probably produce something akin to a pulse at the collector.
The diode prevents the base-emitter junction from becoming too reverse biased (either to prevent base-emitter reverse voltage breakdown or just to "shape" the input signal a little bit).
Best Answer
This is called a Flyback Diode.
Motors tend to have a large inductance.
Once current gets going in an inductor it doesn't want to stop. If you try to stop it anyway by cutting off the current path the voltage will spike to a very high level and probably damage something (the mosfet in this case).
The diode gives the current somewhere to go. It routes it back to the other side of the inductor and allows the current to flow in a loop until it dies down due resistive losses.
The diode is sometimes also called a Freewheeling Diode for this reason.