Read Data verification for prepaid subscribers
If you don't have an ID card issued by the Hungarian authorities, you are asked to visit one of our stores. An ID card issued by another country cannot be verified in the Hungarian register, you have to show your entry or residence documents to our colleagues.
Of course this is also written in an interesting English dialect, but the point is crystal clear: you need to go to a Telekom store to get this verification done. Have your passport with you or Hungarian residence card if you have one.
Activation is the next, relatively easy step which will be done in store. In Hungary typically there's no reason for insane amount of red tape, but in this case there is: organized crime previously bought an insane amount of SIM cards (some reports said 200 000) with the ID of a homeless person and some of that landed in the hand of the terrorists in Paris and Brussels. Opsie. Of course, they had inside help but still, they don't want this to happen again.
Treat it as a genuine Hungarian experience as the poor sods there need to deal with this much red tape all the bloody time.