In The Tempest, Act 1, Scene 2, lines 245-250 when Ariel is enslaved by Prospero, Ariel is to do whatever Prospero orders Ariel to do without any complaint. However, what is the significance of this part?
Remember I have done thee worthy service;
Told thee no lies, made thee no mistakings, served
Without or grudge or grumblings: thou didst promise
To bate me a full year.
A small correction to your question must be noted. Prospero does not enslave Ariel during the course of Act I. Ariel has been enslaved approximately one year at the time that he is introduced to us as Prospero's assistant in his magical mechanization of the weather and the fates of his brother's court. It is Ariel who points out that the arrangement of servitude was for one year. Following the quote you ask about, Prospero takes us through a flashback to tell how and in what condition he found Ariel prompting him to save Ariel before enslaving him.
... Thou best know'st
What torment I did find thee in; ...
... it was mine art,
When I arrived and heard thee, that made gape
The pine and let thee out.
The sense of "bate" as used here is that of to restrain (Random House Dictionary). In this sense (restraining), bate alludes to Ariel's servitude and he uses the allusion to refer to the set length of time of servitude--"bate [restrain] me a full year" and no more than that--while he asks for his freedom. Prospero either counts a year differently than Ariel (perhaps as a year and a day) or has expectations of more than minimal gratitude for the rescue described. In either case, Prosperos declares Ariel will be freed in two days, when Propero's plan is completed.
I will be correspondent to command
And do my spiriting gently.
Do so, and after two days
I will discharge thee.
That's my noble master!
What shall I do? say what; what shall I do?
So the significance of the quote is that Ariel reminds Prospero that he has fulfilled every condition and that Prospero's promise was for one year of servitude but no more than that.