A product in ordinary arithmetic is just a repeated sum: 3×5=3+3+3+3+3. The same goes for the product rule in combinatorics: it is just a repeated application of the sum rule.
I'm going to assume you're happy with the sum rule: this is the idea that if we're counting things (arrangements, outcomes, whatever) that fall under several non-overlapping cases, we count each case separately, then add them up.
The product rule happens when there are many non-overlapping cases that are counted in the exact same way. In that case, you're going to add them all up, and if there are n cases and k outcomes in each case, you're going to get k+k+⋯+kn=nk.
For example, suppose we want to count the number of permutations of the letters A,B,C,D. We split these up into 4 cases: the case where A is first, the case where B is first, the case where C is first, and the case where D is first. Then we notice that each of these cases is identical: in each of them, we have three letters left over, and three slots to place them in. This is a problem you've already solved: there are 3⋅2⋅1=6 permutations of A,B,C, so there are 6 permutations of any three letters. If we have 4cases and 6 permutations in each case, then there are 4⋅6 permutations in total.
A large part of the product rule is just becoming familiar enough with this idea that you do it on autopilot. If you want to know the number of 4-letter sequences from the English alphabet, the argument above suggests that you
- Split them up into 26 cases based on the first letter.
- Realize that each case is identical: counting the number of 3-letter sequences.
- Split up each case into