Indian Railways has numerous monitoring stations. (mainly at railway stations) These monitoring stations update train's location when trains pass 'em.

In between such points, IR may know train's realtime location, but it is not made public.

Update

It is mentioned on the site that there are 6000 monitoring stations. Which means that, on average, there is a monitoring station every 11 kms (66k Km/ 6000). Considering an average speed of 80kmph, the location shown on the app can at max be 8.25 minutes old.

Usually, all railway lines must know the position of every train at any time. This is simply a requirement for avoiding crashes. As far as I know, there are usually pressure/induction sensors at regular intervals that detect train positions. Maybe that is what they are using?

Electric railways usually only supply actually used sections of the railway with power, so they have to know train positions, too.

Indian railways employs a number of methods. The most primitive fallback is the one where the station master updates the system whenever a train passes the station, whether or not it stops there.

http://trainenquiry.com/ already has same information in text/html format. This is just mapped to gui. Personally I prefer html format.

trainenquiry.com is operated by CRIS (which falls under the railways ministry). So they have access to all of IR data.