That's a great question. How much do you want to know about mathematical modelling? There are actually a number of ways of approaching your question depending on how you want to understand it.
To put the Peak into Peak Oil Hubbert picked the
logistic curve. Normally the logistic curve applies where a growth rate is self-limiting. A classic example is population growth. It is a mathematical "feature" of this curve that maximum rate occurs at 50% exploitation. The way I think of it is as the balance point between two opposing forces - growth and depletion.
A mathematical model is chosen for two reasons - the underlying physics matches the assumption of the model, or the model generates a nice fit with the data without knowing exactly why - if the curve fits then that at least suggests the physics has the same properties as the model. I think in the case of Hubbert a bit of both applies.
There are two reasons why the point may not be exactly 50%. Of course, the maths is an ideal case and the real world is messy, that is the model assumes the parameters stay constant throughout, in reality the parameters have some variation.
The other reason is theoretical, that is the model represents the most important factor (growth is self-limited), but does not encapsulate second order features that affect the curve. (It is not obvious how oil extraction is the same as population growth). These secondary features may shift the peak earlier or later than the midpoint of depletion. However the logistic curve will still produce a curve that is close enough.
theoildrum has some excellent articles on a different model which both has a better match with underlying physical assumptions and also produces a better fit to the data. Even the likes of Deffeyes struggle to get their model to fit the data, and solve the problem by saying "the data is faulty", which is poor science, IMHO. The refined models produce more of a tail after the peak.
I hope that explains some of the modelling, if I made any errors it is because I am using a mini-keyboard.

I deliberately avoided getting into the physical properties of oil extraction, in fact I believe it is more to do with economics than geology.