Paṭiccasamuppāda & Saṃyojana

How are the 10 fetters being destroyed?

To breaking the circuit of 10 fetters, the practitioner has to destroy by the continuous understanding the circuit, paṭiccasamuppāda, of 10 fetters as they are.
In breaking of circuit, paṭiloma-paṭiccasamuppāda, Buddha taught to destroy the last fetter, avijjā and taṇhā&upādāna, to break the whole circuit. But most some people can not immediately destroy avijjā and taṇhā&upādāna.
So, in 10 fetters, Buddha describe more explanation of those avijjā and taṇhā&upādāna as 10 fetters, 10 saṃyojana. And Buddha arranged that 10 fetters from the easiest-to-destroy to the hardest-to-destroy. So, someone, who can not destroy avijjā and taṇhā&upādāna at only one time, will try to destroy the easier-to-destroy fetters (such as uddhambhāgiya-saṃyojana) instead, step by step.
By this way, people can more easily destroy avijjā and taṇhā&upādāna, by beginning at destroying 3 orambhāgiya-fetters, which are diṭṭhi-upādāna/sīpabata-upādāna/attvāda-upādāna-paṭiccasamuppāda, =iterations of bhava&vibhava-taṇhā-paṭiccasamuppāda, and ignoring [avijjā] of causes&effects in paṭiccasamuppāda. After they destroyed these 3 fetters, the 7 left fetters will certainly destroy in the future, not over more than 7 rebirth according to Sutta. Aṅ. (3): chakkanipātā, Paṭhama-abhabbaṭṭhānasuttaṃ and the suttas around it.

Can the chain of fetters be seen during vipassana meditation?

Yes, according to Sutta. Saṃ. Ma. Dhammacakkappavattanasutta, the practitioner have to clearly understand the 4 noble truths, which included 10 fetters in the 2nd noble truth, already.