Matrix multiplication logic?

I understand how to compute matrix multiplication and rules of it such as matching the row and columns etc. but I am trying to understand why the final values make sense as vs to the dot product?

Why do you have to multiply both row and column and do the sum to come out a final value in a new matrix, I don’t understand how this math logic work?

So you multiply tensor a*w , w means weight tensor and come out a new tensor, should this be like a dot product?

If anyone with deeper understand of this logic, please share in graph.

I used this tool

Thank you!
Best regards,