The Railcard website you linked is pretty clear.
> For those aged 16-25, save 1/3 off rail fares for days out, seeing family and friends and even festivals!
> For those aged 60 and over, save 1/3 off rail fares for days out, holidays, seeing family and friends, and theatre trips!
So, provide proof of age via ID and get a discount. It's very common on public transit for the young and elderly to get a discount.