In plain words
Ayana means making a journey. When the sun journeys to the north it is called uttarāyaṇa, and when it journeys to the south it is called dakṣiṇāyana.
Uttarāyaṇa is pure, with light, with Śānti, with joy, with fearlessness. Dakṣiṇāyana, on the other hand, is joined with darkness, with unrest, with fear, with bewilderment. The sādhakas who make the journey in dakṣiṇāyana, the sādhakas who must travel that way, perform meritorious works, sacrifices and rites, gifts and charities, and journey on the southern path. Journeying in this way, they reach the moon. The moon is rayi, that is, matter. By material longings, by material desires, by material pleasures, they gain only the material; they cannot gain God. As long as this time of merit lasts they stay in the world of the moon, and as soon as the merit wears away they come back again to the world of mortals. On this very basis: kṣīṇe puṇye martyalokaṃ viśanti. The one who wins an election sits in that assembly for five years. As each year wears away, little by little he moves some distance from the assembly. As soon as the five years are over, he has no seat in that assembly. In the same way, man, by meritorious deeds and sacred deeds, gains fulfilment in the world of the moon, lives there as long as this time of merit lasts, and as soon as this merit wears away he is born again in the world of mortals. One who journeys in uttarāyaṇa reaches the orb of the sun with the feeling of Brahman, with the Jñāna of Brahman, with love for Brahman, with the principle of Brahman; the orb of the sun is prāṇa, and once he has joined this prāṇa, then for him punarjanma na vidyate. He does not come back to this world again. Child! Only when prāṇa is joined with prāṇa does this find its fulfilment. When you join your life to feeling, it turns into nothing but delusion. As long as delusion remains, Brahman will not come to us. So when one attains Brahman through the feeling of Brahman, there is no such thing as rebirth. So, brahmavid brahmaiva bhavati. One who recognises the principle of Brahman becomes Brahman itself. So try to recognise in uttarāyaṇa and dakṣiṇāyana both rebirth and the absence of rebirth.