The great God-realized yogis place great emphasis on vibrations.
What are the vibrations?
Vibrations are some the manifestation of the supreme Sound of God (Aum). This Aum vibration exists at all times and is the seed of creation.
Sri Krishna says in the Gita (8:13) that it is one of the ways to attain the Ultimate Goal.
oḿ ity ekākṣaraḿ brahma
vyāharan mām anusmaran
yaḥ prayāti tyajan dehaḿ
sa yāti paramāḿ gatim
The one who chants the Eternal mono-syllable OM, the vibration of Brahman, constantly remembering Me, leave their body and attain the Supreme Goal.
From this Aum come all other vibrations.
Sri Madhvacharya states in his commentary for Gita verse 17:23:
In all creation His transcendental sound vibration OM is the root of all sounds and all pervasively always vibrating within and without.
This is what Swami Paramhansa Yogananda says about vibrations (Taken from Cosmic Chants by Paramahansa Yogananda):
Sound or vibration is the most powerful force in the universe. Music is a divine art, to be used not only for pleasure but as a path to God-realization. Vibrations resulting from devotional singing lead to attunement with the Cosmic Vibration
The fact that all vibrations link back to the Cosmic Vibration (AUM) gives it their supreme power.
Chanting sincerely naturally attunes one to the Real Self, by cleaning the senses and mind of unnecessary fears and thoughts!
Shri Adi Shankarcharya says this about chanting the Gayatri Mantra:
Sarvabudhi samgnanthah karana prkashakah sarva sakshi prethyagathma iti uccyathe
By praying, chanting the Gayatri, all the inner organs, Gnanedriyas, Mans, Buddhi, Chitta and the Para Brahman in the Form of Witness reveal themselves,the first four being cleaned by the Gayathri.
This is how chanting can be made powerful as said by Sri Yoganandaji (Taken from Cosmic Chants by Paramhansa Yogananda):
Holding the attention unbrokenly on the real Cosmic Vibration, Aum or Om, not on an imaginary or an outward sound, is the beginning of real superconscious chanting.
About how can these vibrations protect us, Yoganandaji says (Taken from Cosmic Chants by Paramhansa Yogananda):
Words that are saturated with sincerity, conviction, faith, and intuition are like highly explosive vibration bombs that have power to remove the rocks of difficulties and to create the change desired.
Basically saying that this superconscious attuning to the Divine Force or AUM causes removal of difficulties.
But how can a vibration remove difficulties or protect or guide or reach God? Sri Ramakrishna says (The Gospel of Sri Ramakrishna, page 370-371 excerpts available):
The sound OM is Brahman. The rishis and sages practiced austerity to realize that Sound-Brahman. After attaining perfection one hears the sound of this eternal Word rising spontaneously from the navel. "'What will you gain', some sages ask, 'by merely hearing this sound?' You hear the roar of the ocean from a distance. By following the roar you can reach the ocean. As long as there is the roar, there must also be the ocean. By following the trail of OM you attain Brahman, of which the Word is the symbol. That Brahman has been described by the Vedas as the ultimate goal."
He is in other words saying it is like a trail. The sound allows you to naturally calm your mind, your restlessness (which is the root of all evil) and make it easy for you to listen to the voice of the Divine.