It's an easy answer and perhaps quite obvious on reflection: the Church knew about each marriage in advance and gave permission. For a fee.

While certain marriages were considered to be against natural law and would never be permitted, such as marriage between a grandparent and grandchild, or parent and child, money and power could get you permission from Rome for many things.

The marriages banned by natural law extended to spiritual family too, a godparent would not be permitted to marry a godchild. This could actually be used to facilitate a divorce an anullment, something that was otherwise hard to obtain. If a parent became the godparent of their own child then they could claim an illegal cosanguinity in their marriage and be granted a nullification.

Henry VIII used the laws of cosanguinity to declare Mary Tudor illegitimate by 'discovering' that he and Catherine of Aragon were in fact related due to her previous marriage to his brother, Arthur. In fact, in order to form this leviratic marriage Henry had already requested and received (read purchased) permission from the Vatican. But Mary Tudor was her father's daughter in many ways. Later, at her own expense, she purchased another leviratic dispensation for her father's marriage and so legitimised herself once again.

This illustrates quite well that a system was in place to permit marriages, and also shows that the system could be manipulated to various ends by those with the power to do so.

Getting back to the Habsburgs... they also purchased leviratic dispensations in order to protect the family lineage and, of course, to avoid diluting power. In their case this was so prolonged and oft-repeated that it caused the end of the Spanish Habsburg line at the end of the 17th century.

The Church, Sanguinity and Trollope, Durey J, 2008

Royal dynasties as human inbreeding laboratories: the Habsburgs, Alvarez G, Ceballos F C, 2013