Wow, didn't know that system!
Why does the system work?
For any number of a day involving only the digits 0,1 or 2 (these are day 01, 02, 03, 10, 11, 12, 20, 21, 22 of the month) there are two possiblities, since both dices have the right digits.
For any other number of a day there is only one possibility: There is exactly one digit which is not a 0, 1 or 2. If it is 3, 4 or 5, you have to use the second dice for it. If it is 6, 7, 8, 9 or 0, you have to use the first dice (not that 6 can also be used as 9).
Note that this system would still work if there are months with 32 days, but not for 33 days any more.
How can we find such a numbering system?
Now we forget about the numbering system and want to find a way put digits on two dice such that every day number 01, 02,..., 31 can be represented.
From the days 11 and 22 we see that both dice must have the digits 1 and 2.
Now there are 8 blank sides left. On one dice there must be a digit 0. Since together with the other dice we can only display at most 7 combinations involving a 0 (using the trick that 6 and 9 are the same), but we need all the combinations 01,02,03,04,05,06,07,08,09, there must be a 0 also on the other dice.
Now there are 6 blank sides left, which we have to use for the remaining digits 3,4,5,6,7,8 (9 is represented by 6). It is easy to see that we are free to distribute those 6 digits to the 2 dice in any way we like (since they only occur in combinations with 0, 1 or 2).