What is the meaning of product in the phrase "Cartesian Product"?
Why we use the word product?
Because the result is very similar to how multiplication works when distributed.
$$a\times b = \left( \underbrace{1+1+\cdots +1}_{a\text{ 1's}} \right)b = \underbrace{1\cdot b+\cdots 1\cdot b}_{1\cdot b\text{ appears }a\text{ times}}$$
This is similar to
$$\{a_1, a_2, \cdots\} \times \{b_1, b_2, \cdots \} = \left\{ \underbrace{(a_1,b_1),(a_1,b_2),\ldots}_{a_1\text{ is on the left for every }b_i}, \underbrace{(a_2,b_1),\ldots}_{a_2\text{ is on the left for every }b_i}, \ldots \right\}$$