Well i think we should break the functionality of the OD function in two aspects.
one being fuel economy and the second being the acceleration
and before addressing both these issues lets understand what actually happens in overdrive mode
in very simple terms, when we turn the overdrive mode ON the automatic transmission is locked to give a specific gear ratio that is lower than 1:1. In explanation i would say it become like the engine input gear is larger than the transmission gear size. (anyone who has seen the gears on bicycles can relate to that very easily)
thus making it less power in and more power out.
Now lets relate this to the acceleration part. When we are running in OD OFF mode the transmission is generally at a gear ratio up to 1:1. Now lets say that we are moving at an RMP of 3000. At this point if we turn the OD mode ON the applied gear ratio would decrease below 1:1 to about 0.6:1. Thus giving power output equivalent to about 4000 RMP and it will make you accelerate at the same RMP, and help you for example to overtake some one quickly
Now move towards economy. Let us suppose we are going on a highway and we are moving at a speed of 100 KPH at 3000 RMP. now if we put OD in to ON mode, we can go to 2000 RMP and be at same speed 100 KPH. Thus the engine will be using less fuel and it will be fuel economical while cruising.
I hope this explanation would have explained to many how and what OD does.
(the ratios or 0.6:1, the RMP equivalence from 3000 to 4000, the RMP chance from 3000 to 2000 for same speed are supposed for explanation and not accurate) (OD gear ratio of different cars is different)