Here is an answer from Ciberdúvidas:
'A designação de átonos e tónicos refere-se à acentuação. As formas átonas são aquelas que estão junto do verbo, designando-se de pronomes clíticos porque dependem fonologicamente da forma verbal, comportando-se como se fossem uma das suas sílabas (ex.: “Ele entrega-se ao trabalho.”).
As formas tónicas têm sílaba tónica e podem estar afastadas do verbo, sendo regidas de preposição no caso de desempenharem a função de complemento (ex: “Para ela, tudo é fácil.”).'
The designation of non-tonic and tonic [stress pronouns] refer to stress (accented syllable). The non-tonic stress ones are ones attached to or next to the verb, and are called "clitics" because they depend phonologically on the form of the verb, behaving as if they were one of its syllables. (For example: Ela entrega-se ao trabalho. She gives herself over to her work.)
[See how it's as if the se were part of the word and the pronoun is unstressed? The verb syllable tre of entrega-se is where the tonic accent falls]
The tonic stress form has a stressed syllable and may not be next to the verb. They are governed by a preposition when they act as a direct/indirect object. (For example, Para ela, tudo é fácil. For her, everything is easy.)
[See how ela contains a stressed syllable in its own right? The e in ela is the tonic stress. Obviously, they can be subjects or objects. You can't append ela to a verb. It can be a subject or an object.]
in Ciberdúvidas da Língua Portuguesa, https://ciberduvidas.iscte-iul.pt/consultorio/perguntas/pronomes-atonos-e-tonicos/8071 [consultado em 01-11-2021]
So, basically, for a learner, is the pronoun stuck to a verb? Then, its pronunciation does not take the tonic stress: "Falar-me dos problemas políticos neste país, me cansa". So, the tonic accent in falar-me falls on the second syllable. Not on the me, where the intonation falls off.
Another way to state this is to say this is that the clitics have falling intonation:
Look at these: (from Priberam)
Ele deu-me um livro.
The tonic stress falls on deu, not on me. So, the stress is falling on me.
Whereas: Ele me deu um livro. has tonic stress on me, it cannot fall off.
Dou-te uma caneta. Same as above.