Commonly this form of definition is from Yaskacharya's Nirukta or Sayanacharya's Veda commentaries. But could not locate it in these seminal works. However a simple examination of the roots will yield मन् and त्राहि --> mind and protect me. I humbly think Hinduism is about being optimal for self and environment drawing positive power and giving back power.(thought, action, word, idea).
The suggestive power of sounds, gestures , symbols , feelings is well known. Mantra is a powerful word channel to gain positivity and lose negativities. Not only suggestion, but repetition and concentration tie down the mind away from fleeting distraction and frivolous thoughts. If you comprehend each word meaning onto it's feeling, it is great. Comprehending the essence or saar is a level higher. Comprehending the Gayatri ( Savitr ) mantra will lift you to a powerful protected state even on one utterance of it.
In the Saman chanting the words are stretched out filling you ears, and then mind and then soul with the divine message.
Below is an excerpt of a brahmachari weeping to a great Swamiji about failure of his mantra siddhi. So mantrahs work differently, as per divine blessing. Try try try.
Trahi, trahi, trahi me.
The Guru is Mahaperiyava Chandrasekharendra Saraswati Swamigal speaking to a brahmachari Sankaran who was disheartened as he failed to connect to the power.