I am not sure if you have, in English, a verb which adds to the sense of "start" the notion of "more or less suddenly", "kinda suddenly". This is the idea behind this construction. By the way, the text itself emphasize this: "Subitamente. Sem por quê (unexpectdly)".

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Note added at 41 minutos (2021-05-07 16:42:20 GMT)
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Similar expressions are "dar a veneta de" (will occur, sudden idea and for no apparent reason (to do something)), "inventar de" (to make a sudden resolution (to do something)), "dar na telha de fulano" (to come unexpectdly, without apparent reason, to the mind of someone).

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Note added at 16 horas (2021-05-08 08:23:20 GMT)
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Another important idea behind this expression "dar de/dar para" is the notion of ​​continuity or habituality in the action that begins (kind of suddenly or unexpectedly or for no apparent reason, or because of an abrupt change in the person's life, etc). For example, in the passage quoted by the asker, the woman begins to have fits of fear that are repeated, regularly, every morning. I did not mention this before because it was already implied in the expression "no meio da manhã", meaning "every midle of the morning".

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Note added at 16 horas (2021-05-08 08:30:46 GMT)
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...meaning, IN THIS CONTEXT, "every middle of the morning".
Jesus, it's very complicated to give an explanation about these things...

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Note added at 1 dia 23 horas (2021-05-09 15:39:00 GMT) Post-grading
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Thank you, AJSComm! As a "bonus material", if you like, have a look at a translation of a very-well known Brazilian song, in which the verbal phrase "dei para", equivalent to "dar de", is used (the translation is available on the Internet, I made some corrections):

Atrás da Porta
Chico Buarque de Holanda

Quando olhaste bem nos olhos meus
E o teu olhar era de adeus
Juro que não acreditei
Eu te estranhei
Me debrucei
Sobre teu corpo e duvidei
E me arrastei e te arranhei
E me agarrei nos teus cabelos
Nos teus pelos
Teu pijama
Nos teus pés
Ao pé da cama
Sem carinho, sem coberta
No tapete atrás da porta
Reclamei baixinho
DEI PRA MALDIZER O NOSSO LAR
Pra sujar teu nome, te humilhar
E me vingar a qualquer preço
Te adorando pelo avesso
Pra mostrar que inda sou tua
Só pra provar que inda sou tua.

Behind the Door
Chico Buarque de Holanda

When you looked deep into my eyes
And your eyes were farewell eyes
I swear I couldn't believe
I didn't recognize you
I bended over
Your body and I doubted
And I dragged myself and I scratched you
And I grasped your hair
Your body hair
Your pajamas
At your feet
By the bed
Without care, without a blanket
On the carpet behind the door
I complained so low
I BEGAN TO CURSE OUR HOME
To speak ill of you, to humiliate you
And to revenge myself at any price
Adoring you inside out
To show you I still belong to you
Just to prove I still belong to you.

A link to a record: https://www.youtube.com/watch?v=W5WG3uOEW30





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Note added at 1 dia 23 horas (2021-05-09 15:51:48 GMT) Post-grading
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So, both "I had started feeling" or "I had begun to feel" are possible, in my view. Maybe the "ing" construction is better, because the idea of continuity is better expressed, I don't know. And I think that even if you don't have in English a verbal phrase like it, in this particular case maybe there's no problem, because the text string somehow completes the meaning: "Subitamente. Sem por quê."