Doctors help injured players rehabilitate faster, with more doctors meaning faster healing. If you employ more than ten doctors there is a chance that your supporter mood will drop at every daily update until you reduce the number to at least ten. You are nonetheless able to use upto a maximum of 255 docs. Each additional doctor has a degressive effect (that means ten docs don't heal twice as fast as five do).
Several sites and helper applications offer tables or calculators to predict how long an injury will take to heal. Here are some popular sites (only the first one is able to give useful results if you want to employ more than ten docs):
This site tells you how long you can expect a player to be injured based on the injury text in the match report.
The best theory (so far) regarding the influence of docs works as follows:
TSI-increase=Const. * (Y+1)^(0.2155)
Y is the number of docs
The formula should be valid, when there are no docs. There is still healing then, so it shouldn't give a 0. If you have no docs, the healing rate would be Const., the natural healing rate (depending on the age of the player in question).
When having e.g. 2 docs the healing rate would be (2+1)^0.2155=3^0.2155=1.27 (times Const. of course). So it increases by 27%.
It is an exponential function, thats what the ^ stands for.