Correct spelling, explanation: this word is an adverb, and as such it ends with the suffix -ly. The adverbial suffix is added to an adjective; in this case to the adjective great. Thus great + -ly is greatly. Grately is incorrect, because the word grate isn’t an adjective so the is no such form as grately and it would make no sense to create such a word, given the meaning of grate (a technique used in the kitchen). Therefore greately is the correct form.

Definition of greatly:
adverb, in a great way
He was greatly amused when he heard the news.
This campaign can greatly improve our company’s image.

Incorrect spelling, explanation: grately is incorrect since grate is a verb so we can’t create an adverb out of it (not to mention it would make absolutely no sense as grate is a way to cut food). Greatly is correct though, because great is an adjective so we can create an adverb by adding the suffix -ly. Therefore greatly is the correct form.

Incorrect spelling

Incorrect spelling

Incorrect spelling