With a mortar and pestle, pound the red chillies, garlic, and palm sugar to make a rough paste. Then add the fish sauce and salt and pound it a bit to mix it together.
Add the papaya and carrot to the mortar and pound a little more to mix the veggies in with the sauce.
Add the lime juice and pound/mix a little more. Then add the tomatoes and peanuts and pound the mixture just a little more to make sure that all ingredients are mixed. Serve and enjoy.
To shred the papaya (and the carrot), there is a special shredder which you can buy (it's available all over Thailand and other countries in Southeast Asia; my guess is that if you looked in specialty shops and/or supermarkets in Canada/America, you could probably also find one). If you can't find a shredder to shred the papaya, though, you could use the old-fashioned method, which is to whack the papaya with a big knife quite a few times, and then pare away the slices you have created to get shredded papaya.