tl;dr
The shawl is red because red is the color of passion. The soul embodied in the human body is trapped by emotion, passion, desire, and action, symbolized by the red shawl.
Deets
In Indian philosophy, red is the color of the passions. Nature (प्रकृति, prakṛti) is said to have three qualities or potentialities (गुण, guṇa):
- purity or goodness (सत्त्व, sattva), associated with the color white
- action or passion (रजस, rajas), associated with the color red
- ignorance or torpor (तमस, tamas), associated with the color black.
The association of sattva, rajas, and tamas with white, red, and black respectively is traditional. I cannot find an Ur-source for this. That would be an entire research project in itself, akin to finding a source for the common belief that the fruit Adam and Eve ate, which Genesis refers to only as the fruit of the tree of knowledge of good and evil, was an apple. But extensive research (i.e., cursory googling) on sattva white rajas red tamas black yields any number of sites that note this color mapping.
These three potentialities are described in the Bhagavad Gīta 14.5 ff. (Mahabharata 6.36.5 ff). Krishna is speaking to Arjuna:
सत्त्वं रजस्तम इति गुणाः प्रकृतिसम्भवाः
निबध्नन्ति महाबाहो देहे देहिनमव्ययम्
तत्र सत्त्वं निर्मलत्वात्प्रकाशकमनामयम्
सुखसङ्गेन बध्नाति ज्ञानसङ्गेन चानघ
रजो रागात्मकं विद्धि तृष्णासङ्गसमुद्भवम्
तन्निबध्नाति कौन्तेय कर्मसङ्गेन देहिनम्
तमस्त्वज्ञानजं विद्धि मोहनं सर्वदेहिनाम्
प्रमादालस्यनिद्राभिस्तन्निबध्नाति भारत
सत्त्वं सुखे सञ्जयति रजः कर्मणि भारत
ज्ञानमावृत्य तु तमः प्रमादे सञ्जयत्युत
sattvaṃ rajastama iti guṇāḥ prakṛtisaṃbhavāḥ
nibadhnanti mahābāho dehe dehinam avyayam
tatra sattvaṃ nirmalatvātprakāśakamanāmayam
sukhasaṅgena badhnāti jñānasaṅgena cānagha
rajo rāgātmakaṃ viddhi tṛṣṇāsaṅgasamudbhavam
tannibadhnāti kaunteya karmasaṅgena dehinam
tamastvajñānajaṃ viddhi mohanaṃ sarvadehinām
pramādālasyanidrābhistannibadhnāti bhārata
sattvaṃ sukhe saṃjayati rajaḥ karmaṇi bhārata
jñānamāvṛtya tu tamaḥ pramāde saṃjayatyuta
sattva, rajas, and tamas, these material aspects bind, O mighty-armed one, the eternal soul to the body.
Amongst these, sattva, purer than the others, is bright and free of sorrow. It binds the soul through a desire for happiness and knowledge, O sinless one.
rajas is passionate in its essence. It arises from desire, and binds the soul through attachment to the results of action, O son of Kunti.
tamas, born of ignorance, deludes all souls through error, lassitude, and sleep, O scion of Bharata.
Sattva is yoked to happiness; rajas to action, O scion of Bharata; tamas, clouding wisdom, to error.
Specifically with regard to rajas, the Maitrāyaṇīya Upaniṣad, roughly contemporary with the Bhagavad Gīta, says:
अंतस्तृष्णा स्नेहो रागो लोभो हिंसा रतिर्दृष्टिव्यापृतत्वमीर्ष्या काममवस्थितत्वं चञ्चलत्वं जिहीर्षार्थोपार्जनं मित्रानुग्रहणं परिग्रहावलम्बोऽनिष्टेष्विन्द्रियार्थेषु
द्विष्टिरिष्टेश्वभिषङ्ग इति राजसान्वितैः (3.5)
aṁtastṛṣṇā sneho rāgo lobho hiṁsā ratirdṛṣṭivyāpṛtatvamīrṣhyā kāmamavasthtatvaṁ chañchalatvaṁ jihīrṣārthoparjanaṁ mitrāṇugrahaṇaṁ parigrahāvalambo'niṣṭeṣvindriyārtheṣu dviṣṭiriṣṭeśvabhiṣaṅga iti rājasānvitaiḥ
Inward thirst, affection, passion, covetousness, violence, love, hate, lying, jealousy, aimless restlessness, seeking the company of friends, family pride, dislike of unappealing objects, attraction to appealing ones, gossip, overspending, these are the results of rajas.
These ideas about sattva, rajas, and tamas circulate pretty freely in popular understandings of Indian philosophy and religion. One needn't have read the Gīta or the Upaniṣads to know that rajas is the active and passionate force that animates human beings, and that it's red—any more than one would actually need to read the Gīta to know roughly what it says about karma. The claim here is not that the lyricist is drawing specifically on these texts. It is just that the lyricist is drawing on well-known religious or philosophical concepts that date back well more than a millennium.
So the body (symbolized by the chadariyā, literally a blanket) is colored red because it's the locus of passion and action. The site linked in the question has a different interpretation that seems entirely wrong to me:
Rang-rej is a person whose job is to dye the material by painting or colouring, and in this song, he symbolises a guru or teacher.
The same site also translates raṅgarej as "master painter". But the word simply means dyer; there is no connotation of mastery. The shawl is red because it's the rājasik (from rajas) force of matter that binds the soul to passion. Notice how the verse about dyeing the shawl red follows immediately after the verse that says people foolishly bespoil the freshly-woven cloth. A freshly-woven, undyed cloth would be sāttvik (from sattva), and so purer than a red-dyed one. And a guru would disentangle the soul from karmic action, not dye the soul red by enmeshing it in worldly desires. So I disagree with the interpretation given at the website.
Notes
- Transliterations from Sanskrit and Hindi are in IAST, except in direct quotations from the site linked in the question that has the song lyric. Renditions of proper names (Krishna, Arjuna) are conventional and ad hoc.
- Translations from Sanskrit are my own, and I am unqualified to translate Sanskrit. I just don't like any of the readily available translations. So use AYOR. Corrections welcomed.
References
The Mahabharata. Book 6, Bheeshma Parva, Chapter 36. Internet Sacred Texts Archive.
The Mahabharata.Book 6, Bheeshma Parva. sanskritdocuments.org.
Maitri (Maitrayani) Upanishad. sanskritdocuments.org