The units are the elements of that have multiplicative inverses. The units of this ring are those that are relatively prime to 6. So 1 and 5 are the only units and they are their own inverses mod 6. What he means by "every number is a multiple of it" is that 5 is a generator of this group under addition.
Then the pattern repeats.
I hope this helps. You may want to try the integers modulo n, or the multiplicative group of integers modulo n