To transport passengers you need to connect two different cities and use a train that drives between these two cities. The train will automatically load passengers and transport them to the other city (if you select your train the number of passengers in this train is indicated by the suitcase-symbol). So all you have to do now is wait until 1000 passengers have been transported between the cities (but you can also connect more cities to your rail-net and use more trains so that you can transport more passengers at the same time). You can also speed up this procedure by using trains that are best suited for 'Express' or 'Passengers' instead of 'Freight'.
Note: The 1000 passengers have to be transported in one game. The achivement is not cumulative over multiple games.... but while building up the infrastructure of your railway empire it actually should pop really fast without further arrangements