Bhashya
त्रयी विद्या हिङ्कारः ।अग्न्यादिसाम्न्न आनन्तर्यं त्रयीविद्याया अग्न्यादिकार्यत्वश्रुतेः ।हिङ्कार प्राथम्यात्सर्वकर्तव्यानाम् ।त्रय इमे लोकास्तत्कार्यत्वादनन्तरा इति प्रस्तावः ।अग्न्यादीनामुद्गीथत्व श्रेष्ठ्यात् ।नक्षत्रादीना प्रतिहृतत्वात्प्रतिहारत्वम् ।सर्पादीना धकारसामान्यान्निधनत्वम् ।एतत्साम नामविशेषाभावात्सामसमुदाय सामशब्द सर्वस्मिन्प्रोतम् ।त्रयीविद्यादि हि सर्वम् ।त्रयीविद्यादिदृष्ट्याहिङ्कारादिसामभक्तय उपास्याः ।अतीतेष्वपि सामोपास-नेषु येषु येषु प्रोत यद्यत्साम, तद्दृष्ट्या तदुपास्यमिति ।कर्माङ्गाना दृष्टिविशेषेणेवाज्यस्य सस्कार्यत्वात् ॥
In plain words
The triple lore is the hiṅkāra. The sequence of sāmans starting with Agni follows because scripture declares them to be products of Agni. Because the hiṅkāra comes first in all ritual acts, and because the three worlds immediately follow as its effects, this is the opening proposition. Agni and the rest hold the role of udgītha by virtue of their pre-eminence; the constellations hold the role of pratihāra because they respond; serpents hold the role of nidhana because they share the sound 'dhā'. Since this chant lacks any other proper name, the term 'sāman' is woven into all as the collective whole. Everything, beginning with the triple Veda, is to be approached in meditation through the perspective of these chant-parts like the hiṅkāra. In all past meditations on the sāman, whatever chant is interwoven into whatever object is to be meditated upon through that exact perspective—just as sacrificial ghee is refined by a specific contemplative attitude applied to ritual parts.
Close to the Sanskrit
The threefold knowledge is the hiṅkāra. The sequential continuity of the sāman starting with Agni is due to scripture stating it as a product of Agni. Because of the priority of the hiṅkāra among all duties. The proposition is that these three worlds are directly following because they are its product. The udgīthaship of Agni and others is due to pre-eminence. The pratihāratva of stars and others is due to being offered back. The nidhanatva of serpents and others is due to similarity with the letter dha. Since this sāman lacks a specific name, the word sāman is woven into everything as the aggregate of sāmans. Everything is like threefold knowledge and so on. The parts of the sāman beginning with hiṅkāra are to be meditated upon with the view of threefold knowledge and others. In past meditations on sāman as well, whatever sāman is woven into whatever, that is to be meditated upon with that view. Just like the purification of ghee through a specific contemplative attitude regarding the limbs of action.