Bhashya
तत् तत्र देहान्तरसंचारे इदं निदर्शनम्- यथा येनप्रकारेण तृणजलायुका तृणजलूका तृणस्य अन्तम् अवसा-नम्, गत्वा प्राप्य, अन्यं तृणान्तरम्, आक्रमम्- आक्रम्यतइत्याक्रमः- तमाक्रमम्, आक्रम्य आश्रित्य, आत्मानम्आत्मनः पूर्वावयवम् उपसंहरति अन्त्यावयवस्थाने; एवमेवअयमात्मा यः प्रकृतः संसारी इदं शरीरं पूर्वोपात्तम्, नि-हत्य स्वप्नं प्रतिपित्सुरिव पातयित्वा अविद्यां गमयित्वा अचे-तनं कृत्वा स्वात्मोपसंहारेण, अन्यम् आक्रमम् तृणान्तरमिवतृणजलूका शरीरान्तरम्, गृहीत्वा प्रसारितया वासनया,आत्मानमुपसंहरति, तत्र आत्मभावमारभते- यथा स्वप्नेदेहान्तरस्थ एव शरीरारम्भदेशे- आरभ्यमाणे देहे जङ्गमेस्थावरे वा ।तत्र च कर्मवशात् करणानि लब्धवृत्तीनि संह-न्यन्ते; बाह्यं च कुशमृत्तिकास्थानीयं शरीरमारभ्यते; तत्रच करणव्यूहमपेक्ष्य वागाद्यनुग्रहाय अग्न्यादिदेवताः संश्र-यन्ते ।एष देहान्तरारम्भविधिः ॥
In plain words
Here is the illustration behind that verse. Picture the grass-leech exactly as described: it reaches the very tip of its blade — that word just means the point where the blade runs out — and takes hold of another one. An 'ākrama' is simply defined as whatever you can step onto. Having secured that new hold, the leech draws its trailing end up to where its front end already is. That is exactly what you are doing at death. You let this old, long-held body fall — the way someone about to slip into a dream lets waking life go — you release the ignorance bound up with it, you drain it of awareness, and by withdrawing your very self from it you reach for a new foothold, the way the leech reaches for new grass, and that foothold is another body. You take hold of it through nothing more than your own stretched-out impressions, draw yourself fully onto it, and right there the sense 'I am this' begins — just as, in a dream, you are already living in another body even while it is still forming, whether that body will move about or stand rooted like a plant. Meanwhile, driven purely by karma, your organs — each already carrying its own capacity — gather themselves together, and outwardly a body starts to take shape from what is, in truth, no finer than grass and clay. And once that array of organs is in place, the presiding powers — Agni and the rest — come to lend speech and the other faculties their support. That, in full, is how a new body begins.
Close to the Sanskrit
That [verse], in this connection regarding the passage from one body to another, is an illustration. That is: in whatever manner the grass-leech — the grass-leech — having gone to, [that is,] reached, the end, [that is,] the termination, of the grass, [steps upon] another blade of grass — an ākrama [is defined as] that which is stepped upon, thus [is] an ākrama — having stepped upon, [that is,] taken as a support, that foothold, draws its own prior part in at the place of [its] final part — exactly so does this self, who is the transmigrator here under discussion, having struck down this body [which was] previously acquired, like one about to enter a dream, having cast [it] down, having dispelled ignorance, having rendered [it] insentient, by the withdrawal of its own self, take another foothold — like another blade of grass to the grass-leech — [namely] another body, having seized [it] by means of its extended impression, draws itself in, and there begins the notion of selfhood — just as, in a dream, one is already situated in another body even while the body, moving or stationary, is being formed at the place where [that] body is to begin. There, moreover, owing to karma, the organs, having [each] obtained their function, are assembled together; and outwardly the body, standing in the place of grass and clay, is initiated; and there, having regard to the assemblage of organs, the deities beginning with Agni resort [to it] for the support of speech and the rest. This is the rule for the commencement of another body.