From the author's README on Thunderstore

Blue Star II over the default skateboard, this needs to be the only skateboard mod enabled otherwise game will crash.

Install Sonic Blue Star II

You need ReSkate and your own copy of skate. on Steam.

Open ReSkateLauncher.exe and go to MODS.
Search for Sonic Blue Star II and click install. The launcher also installs anything it depends on.
Press PLAY. In game, the MODS tab of the ReSkate menu (Insert) turns it on and off.

Or install it by hand: download the .zip and drop it on the launcher window, or unzip it into its own folder in Mods\ beside Skate.exe.

Maps show up in the ReSkate menu's level list. To join a server running a custom map, you need the same map installed. If the game won't load after installing, see known issues.

skatemods doesn't host or review this mod. The download comes straight from Thunderstore. Only install mods you trust, because scripts can run code.