Why should the subthreshold swing value be small?
I am given to understand from the link below that a small value of
  subthreshold swing in MOSFETs implies that there is a better on-off
  current ratio.


Whether or not a better (smaller) subthreshold swing implies a better (higher) on-off current ratio is going to depend on your definition of off-state current. People publishing papers, trying to brag about their amazing on-off current ratio will take the off state current to be the smallest current they can ever measure from their device. In this case, the subthreshold swing does not affect the on-off current ratio. 

Perhaps a more useful off state current would be at some value of \$V_g\$ where you will be operating the device. In this case, an improved subthreshold swing will reduce the current, assuming all other parameters remain constant.


  But, a small subthreshold swing would imply a large subthreshold slope


Correct.


  and hence, at sub threshold values of Vgs, there'll be a larger value
  of current than otherwise.


Incorrect. A higher subthreshold slope means there will be a smaller value of current, for all \$V_g&lt;V_T\$ (until you reach the leakage floor) than otherwise, if all other parameters are constant. 


  Also, why is a better on-off current ratio desirable 


A better on-off current ratio is desirable since, for a given required on-state current, your off-state leakage current will be reduced. In an IC with millions of transistors, every little bit of leakage current adds up and can become significant. This is one of the major limiting factors in modern processor power supply requirements.


  and how does one
  justify that a higher subthreshold swing value leads to an improved
  on-off current ratio?


This again depends on your definiton of off-state current as outlined above. This is only true if you define off-state current to be at a particular value of \$V_g\$.