Learn Russian: Я просто не помню.

The infinitive is помнить (to remember). While you are currently learning verbs with tricky stem changes (like жить or писать), помнить is quite well-behaved! It follows the standard pattern for verbs ending in -ить. You just drop the -ить and add -ю for the "I" form, giving us я помню.

Why is there no word for "do" in the Russian sentence?

Russian does not use helping verbs like "do" or "does" to make a negative sentence. To say "do not", you just place the negative particle не directly in front of the action verb.

Is просто related to the word for "simple"?

Yes! Просто is the adverb form of the adjective простой (simple). In everyday Russian, просто is used constantly to mean "simply" or "just," exactly like it is here.

Can I change the word order to Я не просто помню?

You can, but it completely changes the meaning! In Russian, не negates the word immediately after it. Я не просто помню means "I don't just remember" (implying you also do something else, like deeply understand it). The original Я просто не помню correctly means "I simply don't remember."