Learn English – The saying “Hair of the Dog”

"To bite the dust" means to die or to fail (see e.g. Wiktionary). Picture someone falling down, wounded or dead, quite literally biting the dust (soil, ground, earth). Etymonline says that the first recorded appearance of the phrase is from 1750. The Phrase Finder supplies it as follows:

The earliest citation of the 'bite the dust' version [of the earlier phrase 'lick the dust', from the Bible] is from 1750 by the Scottish author Tobias Smollett , in his Adventures of Gil Blas of Santillane:

"We made two of them bite the dust, and the others betake themselves to flight."

[...]

[Samuel Butler's 19th century translation of Homer's The Iliad] contains a reference to 'bite the dust' in these lines:

"Grant that my sword may pierce the shirt of Hector about his heart, and that full many of his comrades may bite the dust as they fall dying round him."

Whether that can be counted as an 8th century BC origin for 'bite the dust' is open to question and some would say that it was Butler's use of the phrase rather than Homer's.

The origin comes from the Peter tax and the Paul tax:

The expression refers to times before the Reformation when Church taxes had to be paid to St. Paul's church in London and to St. Peter's church in Rome; originally it referred to neglecting the Peter tax in order to have money to pay the Paul tax.

The Peter tax referred to the tax that people had to pay to fund the building of St. Peter's Church, while the Paul tax referred to the tax that the people had to pay to fund the building of St. Paul's Cathedral.

When the idiom says "rob", it takes that if you don't pay a tax, you are robbing a person (pope, in this case) of what that person rightfully deserves.

Thus, the saying means, not paying the Peter tax in order to pay the Paul tax.