In act one, scene two, Ariel explains to Prospero that he created a magnificent tempest, which left the royal party stranded on the island as he was instructed to do. After Ariel elaboearwa on how each individual in the royal party is safe on the island but scattered into groups, Prospero mentions that there is still more work to be done, and Ariel reminds Prospero of their deal. Prospero responds by reminding Ariel of how he rescued him from the wicked witch Sycorax, who confined him in a hollow pine tree for twelve years using powerful magic. Prospero does not particularly blackmail Ariel because he does not demand something from Ariel in exchange for not revealing sensitive, compromising information about the spirit. However, Prospero does threaten to confine Ariel inside a pine tree again for twelve more years if he continues to complain. Prospero's threats depict him as a rather authoritative, insensitive ruler and not the benevolent, compassionate master of the island. He is focused on being restored to his former position as Duke of Milan.
"Blackmail" may not be the most accurate word. Prospero actually bullies and threatens Ariel. He reminds him of how Sycorax kept him imprisoned, forcing Ariel to thank Prospero for the act of releasing him, but then Prospero goes on to threaten Ariel with precisely the same kind of imprisonment if he is disobedient and refuses to do what Prospero orders. Note the threat from Act I scene 2 where Prospero does this:
If thou more murmur'st, I will rend an oak,
And peg thee in his knotty entrails till
Thou hast howled away twelve winters.
This threat is successful (and shocking) because it is precisely the way that Sycorax imprisoned Ariel. The relationship between Prospero and Ariel is one that is of particular interest in this masterful play, as Prospero can be viewed as being far from the benevolent ruler he sometimes tries to be taken for. He can be viewed as exploiting Ariel most cruelly for his own purposes.