OK mate, I'll give you this one, but if you're preparing for an exam you need to start knuckling down to be honest.
In a nutshell - The diac acts like a switch, and isolates the Triac from the Capacitor C1.
At this point the motor will not be running.
As the C1 charges, at some point it will have enough charge to break through the Diac. This is called the ''breakdown voltage'' of the Diac.
So the Diac is now like a closed switch, and allows voltage to go onto the Gate of the Triac.
This switches the Triac on, and allows voltage to the motor.
Adjusting the Variable Resistor changes the length of time it takes to charge up the Capacitor C1.
High resistance set at VR1 means a longer portion of the ac cycle to charge C1. Therefore a shorter period of time for the Diac to conduct; therefore a shorter period of cycle-time that the Triac is getting voltage; therefore shorter period of voltage to motor. So the motor runs slowly.
Reduce VR1 to speed up charge of C1, the Diac conducts for more of the cycle, the Triac is on for longer, the motor gets more voltage and goes faster.
R2 and C2 are called a ''snubber'' circuit. They take away supply voltage spikes, which could trigger the Triac. (They can come from the supply itself, or from the motor as it is switched off).
By the way - your pdf link is calling it an SCR (silicon-controlled rectifier). But there's no SCR in that circuit and you should have spotted that.
An SCR is also called a Thyristor. It can only conduct in one direction. The Triac will conduct in both directions.
So you need to start getting to grips with your component symbols as well ;-)