Correct spelling, explanation: the reason for this spelling is the base form of the adjective simple. Simpler is the comparative form of this adjective. To create a comparative form we add suffix -er to an adjective, and because simple ends with the vowel e we just add -r to the end of the word. The rule is that when the word ends with y this consonant is changed to i when creating a comparative form. This may be the reason for a mistaken form simplier, which is incorrect because simply is an adverb, not the adjective.

Definition of simpler:
adjective, the comparative form of simple, something easier to understand, do, etc. than something else; a more modest, simple, uncomplicated, etc. thing or person than the other
This maths problem is way simpler that that one.
John is a simple man, much simpler than his posh friends.
I will go that way because it’s simpler.

Incorrect spelling, explanation: the form simplier is incorrect because to create a comparative form of the adjective, we add suffix -er or -r to the end of the word. The vowel i is here unnecessary because the base word, simple, ends with e, not with y (which would transform into i, just like in busy → busier). The correct form of this comparative adjective is, thus, simpler.