Here the value of drain current has been given as per the BSIM3v3 MOSFET model, but is there any reference where it has been shown how this value was calculated and what is the meaning of the symbols such as IDSAT0?
You can find a more extensive description of the BSIM3v3 model in this document. There are some formulas for Idsat in this document but I do not know if they can be used with the formula in your question.
In my opinion unless you're a modeling engineer there isn't much need to understand such a model in detail. Also, there are other models like MOS Model 9, 10 and 11 by NXP (they inherited that from Philips) and BSIM4. The EKV model is also an interesting one :-)
In general, for manual calculation you want a simple 1st order model as is described in most textbooks. So I find it a bit strange to grab a formula from BSIM3v3 but not explain the variables used in that formula.