It is the train as such, mr/s bridge-man, that requires to RES-or not-or in other cases its optional. Just compare it to flying : you will also need to RES each sector. In some cases-it varies per country-and if done smartly=i.e. not via that app (see overveiw of seewulf on howtodoit)-connections=same direction onward, may be done for 1 RES. But without exact details no definitive answer can be made.

In most cases this RES is just a mail telling you you are able to sit on (date) in (train nr) from A to B in car X and seat Y. In some cases-notably for these hi-speed trains, there will also be some QR or barcode with it that staff can check or use to open gates.

As you will be travelling with Eurail you don't need to think about the outbound/inbound rule. The outbound/inbound rule (only 2 travel days in your country of residence) only applies to persons travelling with Interrail.