This meaning of "mint" comes from the manufacture of things (particularly coins): http://www.thefreedictionary.com/mint "Mint condition" means "in the same condition as when it was first made", ie. in perfect condition, undamaged and without wear and tear.

Mint means perfect. It is most commonly used when talking about a collectible object, an antique for example. An LP in mint condition would look brand new, no scratches etc. A car in mint condition would look like it has just come out of the showroom.