The line alludes to the common depiction of the god Vishnu as asleep, or at least recumbent, on a many-headed serpent. Here is an example from the Indonesian site ajeg.org. Vishnu is lying on the snake's coils while his consort Lakshmi massages his feet:
Let's unpack the symbology of this depiction and its relevance to Savitri.
The serpent on whom Vishnu lies is called Sheshanaga. शेष sheSha means final, or that which is left after everything else is gone. नाग naaga means, simply, snake. So the snake's name, sheShanaaga, means, roughly, the serpent of the remainder. Sheshanaga represents the endless cycle of creation and destruction. When he loosens his coils, the universe is created; when he winds them, the universe is destroyed. As the efficient cause of universal creation and destruction, Sheshanaga is himself outside this cycle. Hence his name: he remains even when everything else is destroyed.
Vishnu when contained within the coils of Sheshanaga is considered Narayana (नारायण naaraayaNa), the archetypal man or the masculine principle. As such, he represents mind or consciousness as opposed to body or materiality. In Hindu philosophy, matter is considered feminine and active; consciousness, masculine and inert. The word for this material principle, प्रकृति prakR^iti, is also the word for nature. The word for consciousness, पुरुष puruSha, is also the word for man. Sheshanaga, who creates and destroys the material universe through the movements of his body, is prakR^iti. Narayana, who lies inert/asleep/resting on Sheshanaga, is puruSha.
Lakshmi tending to Vishnu represents the complex interplay of the human with the divine. She's the goddess of wealth and material comforts, a paradox in herself. As a goddess, she is consciousness, but as feminine and associated with worldly success, she is matter. She therefore is emblematic of all humans, split as we are between mind and body. Her devoted attention to Vishnu indicates an ethical path, viz., the choice to make matter subservient to consciousness.
So what does this have to do with the lines from Savitri? The lines describe how Aswapati, Savitri's father, has turned his mind away from the material world, the "coil of things" which by its very nature (the pun is deliberate) temporary and limited, and focused it on the limitless, eternal consciousness "asleep" behind it. An answer on Hinduism SE about the significance of Vishnu's resting on Sheshanaga explains why spiritual contemplation is akin to sleep:
Vishnu rests on Sheshnag because it symbolizes that one who has reached the Ultimate Enlightenment, should be in peace and without tension or stress of mind, knowing that Sheshnag, a snake (allegorical to materialistic troubles) is hovering above him.
The word "coils" is well-chosen because it refers both to worries (one meaning of "coil" is turmoil or trouble) and to the snake's looped body.
It's worth noting that in their philosophical significance, masculine and feminine aren't coextensive with male and female. Sheshanaga is a male snake, despite representing the feminine principle of materiality. Lakshmi is female (obviously), but represents the ideal balance between body and mind, female and male, rather than merely a stereotypically sexist gender role. Certainly the kind of slippage one sees between "man" meaning humankind and "person of the male gender" is just as distressingly common with puruSha, with the consequence that women are associated more with the body, men with the mind. But the legend of Savitri shows the limits of such identification.
Savitri is often invoked as the paradigm of the good wife, but as an ethical ideal she is not a weak and submissive woman. Rather, she is a brave woman who follows her own desires: she marries Satyavan and his being cursed to die within a year; she gives up her own lavish wealth to share his poverty; she restores the sight of his blind father (can't get much more symbolic than that); and she outwits Yama, the god of death, to save Satyavan's life. Her own austerities, her concentration on the immaterial, and above all her intelligence, allow her to defeat the god of death himself.
As Gareth Rees has pointed out in an earlier answer, this line also contains an allusion to Hamlet, but the echo is more than a literary flourish. It is a good example of how Aurobindo specifically, and postcolonial Anglophone literatures generally, position themselves firmly within the English literary tradition while yet remaining rooted in their own cultural heritage. Anybody familiar with the English literary tradition knows Hamlet. The allusive line acknowledges and even claims for itself the cultural patrimony of the colonizer. But anybody familiar with Hindu iconography knows the depiction of Vishnu asleep on a snake. So the line participates in an entire philosophical and iconographic tradition unavailable to that colonizer. So much for Kipling's "never the twain", etc. Aurobindo effortlessly bridges the gap between East and West, underscoring the wilful blindness of the colonial project along the way.
The skewering of both sexist and imperialist tropes throughout Savitri is unsurprising, because imperialism itself depended on a patriarchal view that feminized the natives. Colonial tropes represented the colonizer as masculine, cerebral, and civilized, the colonized as effeminate, carnal, and "natural". In other words, they presented the split between the rulers and the ruled as one between the mind and the body. This line's allusion to the complex navigation of the prakR^iti / puruSha distinction in the Indian philosophical tradition exposes the superficiality of this colonialist representation. Just as Aurobindo playfully demolishes sexist tropes, he deftly demolishes imperialist ones too, remaining witty and undaunted like his heroine.
