Actually, the game automatically saves your progress, but it looks like that is broken on the itch.io site despite working fine on both cocoamoss.com and ddrkirby.com. I'll investigate this.
Edit: So, this is due to the way that chrome/safari/etc block cross-domain access to indexeddb in conjunction with how itch.io hosts games in an iframe. I'd need to implement an alternative saving mechanism in order to get this working on itch.io.
In the meantime there are two workarounds:
1) You can enable third-party cookies (in chrome this is called "allow all cookies".
2) You can play the game on cocoamoss.com or ddrkirby.com.