Meaning
To give in / To admit defeat
Example
Depois de muita discussão, ele finalmente deu o braço a torcer e admitiu que eu tinha razão.
After a long argument, he finally gave in and admitted I was right.
Origin
Dar o braço a torcer ("to let one's arm be twisted") means to give in and admit one was wrong, while its negative, não dar o braço a torcer, means to stand firm. The image is bodily: an arm being wrenched stands for being forced to yield, so refusing it means resisting.
Sources: RTP Ensina, Escreva.ai.
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More idioms
Acertar na moscaTo guess correctly / To hit the nail on the headAmigo da onçaA fake/false friendAndar na linhaTo toe the line / To behave properlyArmar um barracoTo cause a public sceneRodar a baianaTo cause a public scene / To go ballisticArrumar sarna pra se coçarTo ask for troubleArrastar as asasTo hit on someone / To flirtAbrir o jogoTo come clean / To open upArregaçar as mangasTo roll up one's sleeves / To get to workAgarrar com unhas e dentesTo hold on for dear life / To seize an opportunity