Direct product versus direct sum

Question

I want to know the rationale behind and the intuition of why they defined the tensor products. I started to read Keith Conrad's handouts. On page 4, paragraph 2 of this handout he says that the direct sum and direct product of two arbitrary modules are the same sets, but the direct sum is a module and the direct product doesn't have a module structure. Why direct product doesn't have a module structure ?? Besides, on the same paragraph he said that linear functions are generalized additions and bilinear functions are generalized multiplications. how come???

Answer 1

I think the author mean that we define an addition on $M\oplus N$, but not on $M\times N$. Moreover we consider $M\times N$ as a generating set for formal sums $(a,b)+(c,d)$.

As to second question I think this expression is not nice and explains nothing.