EcoRoute and EcoChallenge. Fine! But how calculates Gôrmin this?

Your Nuvi obviously knows your speed from the changes in GPS location plotted against time and can calculate acceleration and breaking from the speed rate of change, e.g. :- if you are travelling at 20MPH and 10 seconds later your are travelling at 80MPH, you have accelerated at a rate of 6MPH per second for 10 seconds, the same can be done for breaking. You have to tell the Nuvi what the car's Miles Per Gallon is for both urban and motorway / highway driving, so fuel consumption can also be calculated from distance covered together with acceleration / breaking behaviour.

There is also something called EcoRoute HD, this uses a device that plugs into the car's engine management system (ODB II On Board Diagnostics) connector, this sends info to the Nuvi via Bluetooth, for a more detailed / accurate display and diagnostics, but the calculations you mention can still be done without the 'HD' device. Details of the HD device are here :-
http://jalopnik.com/5440473/garmin-ecoroute-hd-lets-your-gps-silently-judge-your-eco-driving