The colors green, blue, and red representing the ones, tens, and hundreds place values are repeated from previous materials to give the child a sense of mastery.
The beads are the same colors and represent the same amounts as the bead cabinet found in Montessori preschools and early elementary classrooms.
With the checkerboard, the number 12 would be shown with a 1 in the tens place (blue square) and a 2 in the units place (green square).
My daughter is still using other materials to learn her multiplication facts, but I've found that there is no need to wait to perform these larger operations, while learning her facts at the same time. It makes it more fun for her as she feels like she's performing "big kid" work - and she is!
The checkerboard can be used for bigger, more advanced operations too. These multiple digit divisors use what we call the "magic slide".