When I heard Classical Latin for the first time on Horatii carmina quae voce canora Thomas Nudipes pronuntiat, I was surprised to hear what I will describe as changing tones! The reason why I was surprised is because it's the same kind of thing as changing notes such as in barbershop endings such as a downward slide and because I didn't expect to hear this kind of thing in a language! For example: Ibis Liburnis inter: there are a change of tones on the words ferre, firmus, terrā, and perdam. Is there a name for this?
What you seem to be hearing is likely this particular speaker's idea of pitch accent. Pitch accent is a feature of certain languages in which the word accent is not marked by stress (as it is in English) but by tone.
Did Latin really have pitch accent and, if so, what exactly did it sound like? We do not know for certain what the pronunciation of Latin in the classical period was like because, although the Romans left us a great deal of cultural heritage, they did not leave us audio recordings, so we have to rely on linguistic theories and circumstantial evidence. (How we know about ancient pronunciation is discussed in some detail in this question.) Some things we do know with relative certainty, and they are uncontroversial. The question of whether Latin had pitch or stress accent (or both) is not one of these, rather it has been the subject of much debate; I believe this post on Reddit nicely sums it up.
The Dutchman who recorded these poems comes firmly down on the side of pitch accent; he writes so here ("zo goed als zeker door een verschil in toonhoogte tot uitdrukking gebracht" -- almost certainly expressed by a difference in pitch). To back this up, he quotes Cicero, Orator ad Brutum §§ 57 and 58, where, he says, Cicero clearly indicates a pitch accent. In the question linked above, you will see that Sidney Allen's Vox Latina, while older, is still an important authority. Allen dismisses the idea of pitch accent in Latin completely.