The details from the Number Takers, principally of goods wagons but also carriages and locos on occasion, went to the Railway Clearing House and was used for calculating various aspects of sharing revenue, and cost, between different companies. Obviously a junior role, for a 15 year old to be so employed. Mainly at done boundary points between rail companies.
As well as numbers, they also identified damage and thus charging whichever company was responsible for it, following all this by preparing in the office a range of documents. It was essentially a clerical role for those who could write (by no means universal in the 19th century). They normally had their own desk in an office. Very simplistically, it is what the TOPS computer system does on the railway nowadays.