Matrix addition happens columnwise (or row-wise). Placing the columns (or rows) tip-to-tail produces the final result. In order for addition to be possible the matrices must have the same dimensions (both row and cols). Since addition is commutative, we can reorder the matrices without changing the result $$ A + B + C = \Bigg[\ \ A \ \ \Bigg] \begin{bmatrix} | & & | \\ A_1 + B_1 + C_1 & \cdots & A_n + B_n + C_n \\ | & & | \\ \end{bmatrix} $$ We can illustrate this using axes geometry or parallel geometry.
Matrix subtraction is just matrix addition where the second matrix is negated. If we allow the columns (or rows) to "float" as opposed to being attached to the origin the difference between two matrices can be thought of as the vectors going from the second matrix to the first matrix.