Bootstrapping the zero yield curve

I am not trying to create a zero yield curve, but would like to have a better understanding of how the process works when bootstrapping a yield curve with a cubic spline. I understand that you collect diffferent yields from a variety of securities (deposits, eurodollar futures, etc) in order to extract the discounts and then use the bootstrapping method to calculate the prices of securities in ascending order. My confusion stems from how the cubic spline interpolates this information. I do not understand how the following can calculate the zero coupon yields:

a0T3 + b0T2 + c0T + d0 for 0=T1 and so on....

For some reason, these different steps seem very disparate, and I am have a difficult time meshing them together into one process. If someone can please shed some light on this process and how it works (in layman's terms), I would greatly appreciate it.

Also, do fixed income traders know how to construct these yield curves, or are the quants building them for the traders?