What is the reason for strong acids/bases dissociating in water?
The easy answer: 

It comes down to the strength of the bonds. Or that covalent bond strength is directly related to the bond dissociation energy and thus dissociation. 



A little Physical Chemistry:
(The following is how Pauling explained this.)

In physical chemistry, they have something called bond dissociation ($D_o$).

From Atkins (9th Ed 10.4): "bond dissociation energy, $D_o$, the energy required to separate the atoms to infinity or by the well depth $D_e$." - What is meant here by well is a potential well – for now just think of that as the 2D box talked about in physics. 

Now lets shift gears. You may be familiar with bond enthalpy (sometimes just called bond energies). A common formula used in general chemistry is $\Delta H_\mathrm{rxn} = n(\text{bonds broken}) - n(\text{bonds formed})$

It turns out that $\text{bonds broken}$ or $\text{bonds formed}$ is actually referring to the bond energy or bond enthalpy. 

It actually works out that there is a direct relationship to bond dissociation energy and bond enthalpy; which makes sense. It is elegantly related by: 

$\Delta H^o_\mathrm{bond} = D_o + \frac32 RT $



I guess the natural question to ask now is why do some bonds have a greater enthalpy of formation? 

This was not well understood for a long time. In reality there is a lot of factors that can come into play. The way that Pauling initially came up with dissociation energy, $D_o$, in The Nature of the Chemical Bond followed this protocol:


Spectroscopy, (I did this with FTIR for example, but there are several choices) to get a molecular potential energy curve. 
Extract the well depth, $D_e$
Find the dissociation Energy using $D_o = D_e - \frac12 h\nu$


In modern thermo:

Because of the relationship with enthalpy, we can measure enthalpy with calorimetry and back calculate. 

But the best description uses statistical thermodynamics … Which I will not get into.