Resolved: How is outer multiplication computed mathematically?

Hi Carl,
thanks for reaching out! The exterior multiplication in R is something that's not the most typical matrix operation, so I'm glad you're asking this question.
What happens when you use mat.1 %o% mat.2 is that each value of your matrix mat.1 is multiplied by a scalar which is a value of mat.2. And in fact, your mat.1 is multiplied consequtevely with each value of mat.2 and the result is n new matrices, where n = mxl, where m and l are the dimensions of matrix m.
In your example you obtain 4 matrices. Each matrix is the result of multiplying mat.1 with a value from matrix 2. The dimensions you see
" , , 1, 1", " , , 2, 1", " , , 1, 2", " , , 2, 2"
show you which value from mat.2 you multiply with. So 1.1. means the first row and first column of your mat.2 - and the resulting first matrix is mat.1*[1]. 
Then "2,1,: means that you multiply your mat.1 with the value on the second row and first column of mat.2, so mat.1*[2] 
and so on.
For exercise, you can check out these two matrices
A <- matrix(c(10, 8,
              5, 12), ncol = 2, byrow = TRUE)
A

B <- matrix(c(5, 3,
              15, 6), ncol = 2, byrow = TRUE)
B
A %o% B
and let me know if the results of the multiplication make sense.

Best,
365 Eli