Where does that quote come from?

I know of two French pensions, only one that I was aware of dealt with Henry VIII, one that deals with an Edward.

Context for both goes back to when Isabella of France married Edward II of England. Her brother, the French king, died without issue and by English law Isabella and her descendants should be the next heirs. But French didn't recognize matrilineal claims, so in French law, essentially the next closest male relative should be heir. Edward III by virtue of his mother Isabella felt entitled to the French crown, and this was a major factor in the start of the 100 Years War between France and England. Later Henry V at the Battle of Agincourt also won a promise of the French throne but died before he could inherit. English royalty felt slighted twice over for their 'rightful' inheritance.

The on-and-off battles of England and France, and side-parties like the Dukes of Burgundy and Brittany meant that pieces of territory were fought over.

Fast forward to Edward IV, of the War of the Roses, he tried to invade France with the Dukes of Burgundy and Brittany as allies. That didn't go well and King Louis of France said he'd be a better ally and managed to talk Edward down despite at having given safety to 'traitors' to Edward at different points. Louis essentially said, 'hey, you and I can be great allies, I'll pay you a bunch of money, I'm more reliable than the goobs you teamed up with. Also if either of us has rebellions we help each other out. Just leave France and the French throne alone, how about it?" And from that alliance, the Treaty of Picquigny was made which granted Edward IV a yearly pension from the French crown.

I've never heard that it carried on after Edward/Louis death, which doesn't mean it didn't, but I've not come across that. That's the Edward with a pension I'm aware of.

As for Henry VIII, in the Treaty of More, Wolsey negotiated a settlement with the French. England agreed to relinquish some territories that [either they held them, or believed they should hold them] in exchange for a French pension. There were little caveats added but that was essentially it, it was another paying off of England with some alliance and financial settlements attached.