Controlling a 240V AC Universal Motor with Microcontroller, Optoisolator & TRIAC
So first off; I've tried to hook the TRIAC (2N6073AB) to 240 VAC, it scattered into two pieces after about 5 sec.
  


Without a schematic of your wiring we can't say. It sounds as though you mis-wired it or switched it on to a dead-short between mains and neutral. There's a schematic button on the editor toolbar if you wish to update your question.


  
  For the circuit above, I bought 1/4 watt resistors, and I cannot understand how they can handle 240 VAC, seems really strange to me. Or maybe they can't? Like the capacitor in series with the 39 ohm resistor, how is that even possible?
  




Figure 1. Carbon film resistor with exposed carbon spiral (Tesla TR-212 1 kΩ). Source: Wikipedia Resistor.

There are three main specifications to watch when using resistors:


The resistance value. This is obvious.
The voltage rating. In Figure 1 we can see a spiral resistance track around a ceramic core. The track is probably about 20 to 25 mm long if unwound. Above a certain voltage electrical breakdown will occur across the resistor - perhaps jumping between turns on the spiral. Typically they're good to 200 to 250 V but mains voltage can peak at \$ \sqrt {2} V_{RMS}\$ so when we use them on 230 V mains we generally use two in series.
The power dissipation has to be kept below the rating of the resistor. This can be checked using \$ P = \frac {V^2}{R} \$ or \$ P = I^2R \$.


It may help to consider capacitors as two layers of foil separated by an insulating film. All that is required is to make the insulation layer thick enough to withstand the applied voltage. 


  
  What role does the 330 ohm resistor play, why is this one needed? And it also says "for highly inductive loads, change this value to 360 ohms", what value? Is it the 39 ohm resistor? And why change it, is because of the high start current for motors?
  


There are three resistors in your circuit.


The 39 Ω and 0.01 µF capacitors form a snubber which protects the triac against sudden changes in voltage. You don't have to worry much about this but see Littlefuse Phase control using thyristors for more information.
The 360 Ω resistor choice is covered in my answer to How is the gate trigger resistor value calculated for a triac.
The 330 Ω resistor provides a neutral return path for the zero-crossing detector. It is explained in my answer to Using AC current to trigger Triac.



  
  For the 0.01 microFarrad capacitor, where does this value come from? And from what I've softly read, the snubber circuit is to prevent the phase shift between voltage and current caused by the motor, right? Will these capacitors do: Blue Ceramic Disc Capacitors 1KV 1000V 103PF 0.01uF?
  


Covered above. I don't know what the 103PF means. The Littlefuse article should give you enough detail on this.


  
  "The output of most microcomputer input/output (I/O) ports is a TTL signal capable of driving several TTL gates. This is insufficient to drive a zero-crossing TRIAC driver."
  


You don't quote a source for this but it looks a bit out of date. Most of the micros can now switch 20 mA and this is plenty for an opto-isolator LED.


  
  I guess that's not the case with Arduino Nano since it uses PWM signal? Or do I still need the NAND-gate? And if someone would like to explain why the zero-crossing TRIAC driver doesn't accept certain signals, I would be grateful.
  


PWM isn't used with triac control circuits. This is explained in my answer to Activating SSR for an AC motor via PWM input.

Further reading:

ON Semiconductor's 240 page Thyristor Theory and Design Considerations Handbook is a very in-depth look at the topic but is fairly readable if you pick an aspect of interest.