Are you talking about
http://super4i.com/r4i-download-e.html specifically, or is the official website on your packaging/cartridge label different?
Is your DSi brand new or fully updated (1.4 Firmware in the System settings)?
Though it claims to be compatible with all types of DS/DSi, the product's initial release was before the major flashcart-blocking DSi Firmware 1.4 was introduced. The site claims in poor English that this product (made) after Aug 30 of last year is compatible with 1.4 Firmware. This implies that if yours was made before that it's never going to work as they don't have a flash update for the firmware of the cartridge.
Sometimes the Kernel / Operating Software is called firmware, but the Kernel alone will never change the cartridge firmware; a flash update for flashcarts is typically a small homebrew that is run in the Kernel GUI from the MicroSD that modifies the cartridge's firmware.
The most recent Kernel release is from the end of last October, which makes it pretty safe to say they've given up and bailed on supporting the product's future.
Like it seems for you, I too started out on this site looking for solutions to problems I was having with an "R4i" variation of R4-clone. After a lot of research, and before I actually became a GBAtemp member, I decided the solution was to get an Acekard 2i. Since then, the Supercard DSTwo has been released as well. They seem to be the two most recommended flashcarts for use with DSi 1.4 and beyond because they can be internally updated. Acekard 2i has a very wide knowledgebase and Supercard DSTwo has a pretty active official development team.