"Spise" is the infinitive form, and belongs together with the infinitive mark "å". "Å spise" = to eat (most infinitives ends in e).
"Spiser" is the present tense (also very common to get the present tense by adding an r), which in English could be either "eat" or "eating", there is just the one in Norwegian.
And when you get to stuff like conditional forms, they are formed by some helper verb + infinitive form.
And we have a sort of continuous present form in Norwegian. If I get a call while eating dinner, I'll probably respond "jeg sitter og spiser", which literally translates to "I am sitting and eating" but in effect just means "I'm eating".
"I'll eat whatever/anything I want/wish" comes out like "jeg spiser det jeg har lyst på". Which is a slightly different sentiment from "jeg spiser hva som helst". If you'll eat "anything/whatever" you really don't care, you just like chewing. As soon as there are wanting and wishing, you do mind, and you fall out of the scope of the Norwegian original.
The accepted answers are just "I'll eat anything/whatever" and "I eat anything/whatever".