Your understanding of the GPL is a bit off. GPLed code doesn't generally force you to GPL the output of the program, but that's just because the output of the program is normally not a derivative work of the program. For instance, if you write a novel in Emacs, your novel is an entirely original work. The fact that you used Emacs as a tool doesn't mean that your novel is somehow based on Emacs.
This generator is different. It generates sprites by putting together art assets made by the program's creator. That means that the output contains a substantial amount of content made by the program creator, which means that it would likely be covered by the program creator's copyright. If the content was licensed to you under the GPL, you normally have to abide by GPL restrictions when you redistribute the content or incorporate it into your own program.
Copyright holders can, if they want, grant additional permissions to their content and allow you to incorporate that content into nonfree software. For instance, the GCC compilers have an additional permission for their runtime libraries to let you use them in any GCC-compiled code; without that permission, GCC could only be used to compile GPL-compatible code. But unless the copyright holder has granted those permissions, when program output contains large amounts of their copyrighted content you're bound by the GPL when using it.
In this case, the program specifically says that the art is dual-licensed under GPLv3 and CC-BY-SA-3.0. Since the generator's output is a combination of premade art assets (the buttons just let you tell the program which art assets to combine), the output is a derivative work of those assets. You can't use it unless your use is compatible with the GPL v3 or CC-BY-SA-3.0 (either works, since dual licensing lets you pick either license).