It is not so much the wound that Faramir received in battle that brings him close to death. Aragorn reaches the heart of the matter when he says to Imrahil, “Weariness, grief for his father’s mood, a wound, and over all the Black Breath”. All these things have finally overcome the valiant Faramir. All his life he has resisted the creeping shadow both in the rise of Mordor beyond the borders of Gondor and within the hearts of his own people and now, at last, his hope is gone.
It is not by Athelas alone that Aragorn heals Faramir. Tolkien does not enter into any explanation of the process but simply describes what Aragorn does.
“Now Aragorn knelt beside Faramir, and held a hand upon his brow. And those who watched him felt that some great struggle was going on. For Aragorn’s face grew grey with weariness; and ever and anon he called the name of Faramir, but each time more faintly to their hearing, as if Aragorn himself was removed from them, and walked in some dark vale, calling for one who is lost.”
What Tolkien describes here is some form of the coinherence about which the Inklings used to speak and an idea which was introduced to them by Charles Williams. Williams believed that Christians could voluntarily bear the suffering or burden of another and so aid their healing. Aragorn’s apparent journey away from himself and his profound weariness as he makes this journey seems to suggest that this is what is happening. For those who would like to explore this idea further I would warmly recommend the work of Sørina Higgins on Charles Williams which you can explore by going to https://theoddestinkling.wordpress.com and clicking on coinherence in the tags on the right hand side of the page.
It may be that Aragorn is able to call Faramir back from his journey towards death by this means but the healing is made complete when Bergil arrives with athelas. Aragorn crushes two leaves and casts them into a bowl of water and life is restored to both the healer and the one who is near to death.
“The fragrance that came to each was like a memory of dewy mornings of unshadowed sun in some land of which the fair world in Spring is itself but a fleeting memory.”
As you read the account of the healings in this beautiful chapter you will note that the fragrance of athelas is somehow different for each person that is healed. It is a beautiful expression of the unique relationship between the one who is hurt, the means of their healing and the healer. Surely in Faramir’s case we catch a glimpse, just for a moment, of his deepest yearning. When Faramir explained to Frodo the meaning of the ceremony that he and his men observed before eating in Henneth Anûn he spoke of his longing for the restoring of Gondor and also for something deeper even than that longing. He spoke of “that which is beyond Elvenhome and will ever be”. https://stephencwinter.com/2015/09/08/faramir-remembers-that-which-is-beyond-elvenhome-and-will-ever-be/
Faramir has long pondered that which Númenor and even Valinor can only point to. He is one who cannot stay at the surface of things and so passes through his experience as son of the Steward of Gondor through the history of his people and unto their origins in Númenor. And on arriving there and pondering both its glory and its fall under the shadow he goes deeper yet until he comes to Valinor which is forever closed to them. He will know that it is at the surface of Valinor the deathless land that the corrupted kings of Númenor stayed and so desired to possess it and the gift of immortality and so he passes deeper yet to what lies beyond Elvenhome. This is what he and all in the Houses of Healing glimpse just for a moment. It is a glimpse into the most secret place within his soul, into his most true self, even into the deepest reality of all and so he is called back from the shadows into light and life and into service of the king for whose return he has long waited.