Michaelis-Menten equation type - how it is derived?
These kind of equations (the Michaelis-Menten [MM] like term) denote  saturation kinetics. The basic mechanistic assumption behind saturation kinetics is this:

A rate (of lets say product formation) is dependent on the concentration of a molecule such that the rate increases linearly with increase in the concentration of the molecule.


Example 1: Substrate in an enzyme catalysed reaction
Example 2: Transcription factor in a transcriptional activation


However at a certain point increasing the concentration of the molecule will have no effect on the rate of reaction. For the above examples this point will arrive when (respectively):


Active sites in all the enzyme molecules are bound and there is no more space to accommodate a new substrate molecule.
All the binding sites in the DNA are occupied by the existing TF molecules and no more can bind.


In other words, the reaction kinetics changes from first order to zeroth order.

We can provide mechanistic explanations for some cases but in many we do not know how exactly the saturation happens. In modelling we just use a function that will mimic this kind of behaviour.


  Why functions of the form $\dfrac{X}{K+X}$ ? 


In deterministic modelling using differential equations, it would be preferred to use a continuous function (instead of using a piecewise linear function) because:


Numerical integration is better in case of continuous functions
In biochemical systems, operations are always continuous. There is no digital logic.


The MM function can actually be derived using an equilibrium or a quasi-steady state approximation (See the wikipedia page for the explanation).

In the MM function, when $X \gg K$ then the function is almost 1. This multiplied by the maximum rate constant will give you the max catalytic rate. At $X=0$ the rate will also be equal to zero. Note that in this model the catalytic rate will not indefinitely increase with $X$.

There are several functions that have values between 0 and 1 for increasing concentration of the substrate. Hill function and logistic function are common examples. However MM is a simpler model. You can always revise your modelling approach when you have better information of the biological system.

Now back to your question:

In this model $C$ denotes alcohol concentration $\alpha$ denotes maximum clearance rate and $\beta$ denotes half saturation concentration (concentration of alcohol at which the rate of clearance is half the maximum rate). Intuitively you can think about the mechanism like this: rate of clearance of alcohol depends on the level of alcohol i.e. if you have a lot of alcohol the amount of it removed per unit time will be higher. Think of it as water that is pumped through a pipe; more water in the tank will result in higher flow rate. However when the kidney (or pipe) is overloaded or "clogged" with an already excess of alcohol then clearance rate would remain the same (the maximum that the kidneys can excrete). 


  the amount of alcohol that is introduced in the body, equals the amount of the alcohol that leave the body


That is called mass balance which can be summarized as follows:

Accumulation = Input + Production - (Output + Degradation)