It's dangerous to endorse someone's intuition because it may be wrong in some cases and different people may use different intuition – sometimes correct, sometimes incorrect.
The force is a quantity $F$ that tells us how strong is a cause of the mechanical change of some objects or their motion. How the objects actually change – or how their motion changes – when the force is applied is a different question that requires another calculation or reasoning on top of the knowledge of the value of the force.
When a large car collides with a small car, they are exerting the same forces of the opposite direction (Newton's third law) on one another. But that doesn't mean that both cars will be "equally affected". If the effect of the force is to change the speed of the cars, the lighter car will be affected more by a given force because $F=ma$.
Similarly, a force is something that may tear a thread to pieces. But the same force $F$ may be enough to tear a thin thread but not a thick one. Each thread has its own value of $F$ at which it (probably) tears apart.
You may visualize every force as being "equivalent" to the force obtained by hanging a weight of mass $m$ somewhere. This creates the force $F=mg$ in the gravitational field $g$. If one is vague enough, one could confuse force with "the distance by which a car gets kicked" or "the momentum" or "the energy" or dozens of other related quantities. All these quantities are completely different and it is indeed a terrible beginner's mistake to confuse any pair of these. But unless one uses some precision and mathematical expressions – and it seems that the OP has avoided them – it is very likely that one will confuse pretty much everything with everything else.
Composite units are no different from the units of areas. The area "one squared meter" is "one meter" times "one meter" and one may literally visualize "one squared meter" as the are of the square with both sides equal to "one meter". The volume is totally analogously the product of three distances and the "cubic meter" may be defined as the volume of the cube with the "one meter" sides in all three independent directions. All these things are rooted in the fact that the area of a rectangle is $A=L_x L_y$, the product of two distances (or three, in the case of volume).
The other composite (product) units are analogous in the mathematical sense. One Newton is one kilogram times meter over squared second (mass times acceleration). If one multiplies the mass $m$ and acceleration $a$, one simply has to get units that are products of the units used for $m$ times the unit used for $a$. That's the case for the very same reason why the squared meters are units of areas.
Imagine that we decide that the distance we know as "one meter" will be called "two meters" because we redefine meters. Because the areas are $A=L_x L_y$, this means that "one squared meter" will be called "four squared meters" because this square may be divided to $2\times 2 = 4$ "new squared meters". So it's important to remember the powers of the base units because different quantities would be affected differently if the meaning of the base units changed. For areas and volumes, these things are particularly self-evident and may be honestly visualized. For all other composite units, the visualization is a bit abstract but the mathematical reason why the units must be multiplied correctly is always the same.
