Correct spelling, explanation: greatly is an adverb, so it ends with the suffix -ly, which is added to an adjective. This is how we create adverbs. Here, the adjective is great, so greatly is the correct adverb. Grate however is a verb, so we can’t make an adverb out of it, not to mention that it would be nonsense to do so.

Definition of greatly:
adverb, largely, in a great way
This improvement will greatly minimize the danger of the plan.
The pay rise greatly improved employees’ trust.

Incorrect spelling, explanation: this form is incorrect because the base word here isn’t grate but great – greatly (grate is, by the way, a verb, so we can’t create an adverb out of it). To create an adverb from an adjective we add -ly to it, so grately is incorrect, but the correct form is greatly.

Incorrect spelling

Incorrect spelling

Incorrect spelling