I'm not sure this is the information you're looking for, but the word "unreserved" in that quote is critical. Although reserved seats are sometimes optional (on some trains in Germany and the UK, for example), this is not typically a choice for the passenger. More commonly a train is either reservations-mandatory or reservations-not-possible.
The slower trains (often referred to as "regional" here) have no option to reserve a seat, and tickets don't usually tie you to a single departure. I believe regional-train tickets are usually date-specific. In France there are peak and off-peak fares, so that's a restriction to be aware of. And in the UK--in addition to the off-peak option--some origin/destination pairs are served by multiple rail companies, in which case a ticket may be restricted to a specific company.
In some countries (including Italy) you must validate regional tickets not purchased online before getting on the train. That consists of sticking the ticket in a machine before boarding. The machine prints the date and time on the ticket. You then have X hours (or all day) to get to your destination. That may allow time to hop off and make a quick visit to a town along the way. In some countries a conductor comes through the train, punching the tickets; I don't know what impact that would have if you wanted to hop off and back on.
For some of the fastest trains (Frecce in Italy, TGVs in France, etc.) a seat reservation is always required. There are some categories of trains on which it can go either way; the InterCity trains vary, I think. On the Frecce, etc., obviously a seat reservation would be meaningless if the train departure time were not also specified. So for those reserved tickets, you cannot hop off and back on. Or, more precisely, you can hop off but will have to buy a new ticket before hopping on another train to continue your trip.
Incidentally, when traveling in Italy to or from a smaller city not served by the Frecce trains, it really pays to look at the schedule details--particularly the total elapsed time--before selecting a departure time. If part of the trip has to be on a regional train, you will often have a choice between a cheaper all-regional ticket with some time flexibility and a more expensive combination of regional and Freccia trains. The regional+Freccia combination may not necessarily be significantly faster, because hanging around at the transfer point may eat up all the time you saved flying through the countryside on the Freccia leg of the trip. And the regional+Freccia combination will tie you to the specific Freccia train you selected.