Do not confuse them with Spike Shoes, which have a similar appearance.
Spring Shoes increase the height of your jumps by one tile, allowing you to jump further and grab ledges that were previously out of reach. Running jumps will go even higher. They work great in tandem with climbing gloves and a cape, but are made redundant by the jetpack.
One caveat to wearing spring shoes: it can be harder to hit aerial enemies (such as Spiders and Bats) with your whip while using them, as the different jump speed takes getting used to and can throw off your timing.
Spring shoes can be worn at the same time as spike shoes, allowing for both effects at once.