The most intuitive way of thinking of it seems to be to picture two objects each made up of 100 "particles" (imagine indivisible points of equivalent and constant mass) at a given separation $d$. We can build up to this picture and see why the product of masses makes sense.
First, imagine the simplest picture: 2 particles, $A$ and $B$ at separation $d$. Let's say that $A$ experiences 1 unit of force due to $B$ and $B$ experiences one unit of force due to $A$. Now let's add another particle $C$ that is right next to $B$. The gravitational attraction between $B$ and $C$ is very large but we can imagine it being canceled by some repulsive force of the particles touching. Now $A$ experiences 2 units of force, 1 from $B$ and 1 from $C$. The system of $B$ and $C$ together experiences 2 units of force, 1 from $B$ with $A$ and 1 from $C$ with $A$. So we can see how each particle in a system independently experiences gravitational attraction from each individual particle in the other system.
Now imagine the two objects ($X$ and $Y$), each with 100 particles stuck together, and the two objects at separation $d$. Now each particle in $X$ experiences attraction from every particle in $Y$ (and vice versa). So 100 partices in $X$ each individual experience 100 separate attractions, each of 1 unit of force. Therefore the total attraction is $100\times100=10000$ units of force.
Finally imagine $X$ has 50 particles and $Y$ has 150. The total mass of the two objects is the same as the previous case, but by the same argument $X$ has 50 particles which each independently experience attraction with 150 particles (with each instance of attraction imparting 1 unit of force), therefore $X$ experiences $150\times50=7500$ units of force.
Hopefully it is now more clear why force goes with the product of masses.