How is it that Force = Mass $\times$ Length / Time ^2? I understand how $F=ma$ but what I am looking for is a diagram, idiom or concept that explains how force can be explained (in a partial layman's terms) as a combination of the dimensions; Length, Mass and Time.
 A: Distance travelled can be measured as a length (meters) [$m$].
Now if we differentiate this function with respect to time ie. get the change in distance divided by the change in time we get the average speed which can be measured in distance divided by time (meters per second) [$m s^{-1}$].
If we repeat this process again we can get the acceleration which is the change in speed divided by the time taken, (meters per second divided by seconds) [$m s^{-2}$].
In order to accelerate an object we have to do something to it (apply a force). This force needs to be bigger for more massive objects (mass measured in $kg$).
We define a force of one newton [$1N$] to be the thing that we have to do to an object that weighs one kilogramme [$1kg$] to accelerate it such that it increase its speed by one meter per second every second [$1ms^{-2}$].
Thus $1N = 1 kgm s^{-2}$
Force = mass times length divided by seconds squared.
A: Actually $$\tag{1}\sum_i\vec{F_i} = \frac{\mathrm{d}}{\mathrm{d}t}\vec{p}$$ is the so called "Newton's second law". It's an axiom if you will, the validity of which must be tested by experiments. The law sometimes (if the mass of the body under consideration is constant etc) simplifies to (say in the $x$-direction)
$$\tag{2}F_x = ma_x. $$
Now, to avoid confusion notice that I use cursive/italic letters e.g. $"m"$ which stands for the mass of some particle while I use ordinary roman "m" which is short for the "meter". Don't mix these. Also when discussing the [dimension] of a quantity (e.g. length or time) one writes these with capital upright letters such as $\mathrm{L}$ and $\mathrm{T}$. 
Now, mass has an SI-unit of 1 kg which is an SI base unit, and acceleration has a derived unit of $1 \mathrm{m/s^2}$. The corresponding dimensions are Mass ($\mathrm{M}$) and Length per Time per Time or $\mathrm{L/T^2}$ respectively. Hence the dimension of force is just the product of these
$$\tag{3}[F] = [ma] = [m][a] = \mathrm{M}\cdot \mathrm{L/T^2}$$
which translates to the unit of force $\mathrm{kg\cdot m /s^2}$ AKA 1 Newton or $1\mathrm{N}$ in SI-units. 
Another example is the $p$ of equation $(1)$, which stands for momentum which basically is mass times velocity so $p = m\cdot v$. We could then ask what the unit of $p$ is? Well it is the unit of mass times the unit of velocity (which in SI-units is $\mathrm{m/s}$ but of course you might use another unit if you like). Then the unit of $p$ is (in SI-units) $$\mathrm{kg~m/s}.$$
What is the dimension of $p$? Well it is 
$$\tag{4}[p] = [mv] = [m][v] = \mathrm{M~L/T}. $$

Appendix: 
Notice that in equation $(1)$ you've got a $\mathrm{d}t$ "downstairs" which then gives you another $\mathrm{T}$ downstairs in $(4)$ hence giving you $(3).$ So the dimensions of the LHS and the RHS of equation $(1)$ are (as they should be) equal to one another. 
A: Imagine you are given a large number of different springs, and a large number different objects. You attached different objects to different springs (with the same extension for simplicity). Repeat the experiments many many times with different combination, and measure the accelerations $a$ of the different objects when they are attached to different springs. After a lot of experiments, you find a law. The law is that you can assign to every spring a number called "force" $F$ exerted by the spring, and to every object a number called the "mass" $m$ of the object, so that the acceleration $a$ is always given by $F/m$. Therefore $F=ma$. To remind people that the number $m$ is referring to this property of the object, you add "kg" after the number. For similar reason, acceleration $a$ is described by a number followed by $\text{m/s}^2$. Therefore, [Force] = kg m/s$^2$.
