In plain words
Just as a person casts off worn-out clothes and takes up new ones, so the embodied one casts off worn-out bodies and enters others that are new.
Word for word
As a man, casting off worn-out garments, takes other, new ones, so the embodied [one], casting off worn-out bodies, passes into other, new ones.
वासांसि जीर्णानि यथा विहायनवानि गृह्णाति नरोऽपराणितथा शरीराणि विहाय जीर्णानिअन्यानि संयाति नवानि देही
Other commentaries
The commentators
▸Bhagavadgītābhāṣyam
प्रकृतं तु वक्ष्यामः । तत्र आत्मनः अविनाशित्वं प्रतिज्ञातम् । तत् किमिवेति, उच्यते —
वासांसि वस्त्राणि जीर्णानि दुर्बलतां गतानि यथा लोके विहाय परित्यज्य नवानि अभिनवानि गृह्णाति उपादत्ते नरः पुरुषः अपराणि अन्यानि, तथा तद्वदेव शरीराणि विहाय जीर्णानि अन्यानि संयाति सङ्गच्छति नवानि देही आत्मा पुरुषवत् अविक्रिय एवेत्यर्थः ॥ २२ ॥
In plain words
Now back to the matter at hand. There the claim was made that the Self is indestructible. Like what? Here is the answer:
'Garments' means clothes. 'Worn out' means gone thin with use. Just as, here in the world, a man throws such clothes away — 'having cast off' means abandoning them completely — and 'takes' new ones, meaning he puts on ones not yet worn, 'other' ones, meaning different from the ones he dropped: in exactly that way the embodied one — and 'the embodied one' means the Self — casts off worn-out bodies and 'goes to' other, new ones, that is, comes together with them. Like the man in the example, it is changeless. That is the meaning.
Close to the Sanskrit
But we shall speak of the topic at hand. There the indestructibility of the Self was asserted. As to 'That is like what?' — it is said:
'Garments' — clothes; 'worn out' — gone to feebleness; just as, in the world, 'having cast off' — having wholly abandoned them — 'new' — brand-new ones — 'he takes' — he receives; 'a man' — a person; 'others' — different ones: so, in exactly that way, having cast off worn-out bodies, the 'embodied one' — the Self — 'goes to' — comes together with — other, new ones; like the person in the example, it is changeless indeed. This is the meaning.