Through your question, you are asking why the Gods Indra, Surya, Garuda and Brihaspati are referred in the swastik mantra.
स्वस्ति न इन्द्रो वृद्धश्रवाः स्वस्ति नः पूषा विश्व्वेदाः ।
स्वस्ति नस्तार्क्ष्यो अरिष्टनेमिः स्वस्ति नो बृहस्पतिर्दधातु ॥
Before answering the question, I will clarify that the above mantra is a swastik mantra.
The meaning of the mantra given by you is -
May Indra of great fame cause welfare to us. May the omniscient Pusha (sun god) cause welfare to us. May Garuda, the remover of all inauspicious things cause welfare to us. May Bruhaspati also cause welfare to us.
So, my explanation is
In the Vedas, Indra is the king of Svarga (Heaven) along with his capital city Amaravati and the Devas. He is the deity of the heavens, lightning, thunder, storms, rains, river flows, and war. He is celebrated for his powers, and the one who kills the great symbolic evil/ asura named Vritra who obstructed human prosperity and happiness. Indra is seen as the storm god releases the rains nourishing the parched land, crops and thus humanity.
In the Vedas, Surya is the solar Deity representing Sun. Surya is an important heavenly body appearing in various Indian astronomical texts. He is considered the creator of the universe and the source of all life and the supreme soul who brings light and warmth to the world.
Garuda is described as the king of birds and a kite-like figure.
Garuda is a legendary bird and is the vehicle mount (vahana) of the Hindu god Vishnu. Garuda represents birth and heaven, and is the enemy of all snakes/evil.
In ancient Hindu literature Brihaspati is a Vedic era sage who counsels the gods, while in some medieval texts the word refers to the largest planet of the solar system Jupiter. His knowledge and character is revered, and he is considered Guru (teacher) by all the Devas
So, as I wrote above these Gods together represent everything and are the Supreme souls. So, by this Swastik mantra we pray to them for enlighting us and to protect us.
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