how does the acid act in removing rust?

Answer 1

Phosphoric acid may be used by direct application to rusted iron, steel tools or surfaces to convert iron(III) oxide (rust) to a water soluble phosphate compound.

There are a few products on the market which fall into the category of organic rust converters. These products contain acid to convert rust to black oxide and polymers which bond to rust.
Another technique for removing rust is etching with Phosphoric Acid. Phosphoric Acid has a unique property of dissolving iron oxide quickly while etching iron very slowly.
Oxalic acid opeartes to remove rust by forming a water-solable complex ion (called a chelate) around each iron ion.
All acids contribute some hydrogen to the metal structure.....

Answer 2

Rust is a base - adding an acid converts into a soluble salt, which can be washed away.