From which act in "The Tempest" does the quote below come, and what are the circumstances in which it is spoken?But this rough magic I here abjure;...

It is from Act Five, Scene One of The Tempest and occurs when Prospero, a magician, decides to abandon his magic potions and his communion with magic powers and return to a common humanity. Often identified with Shakespeare at career’s end, an old man with flowing robes and beard, bidding adieu to his books and magic and to “ye elves of hills, brooks, standing lakes, and groves” (5.1.33), with a fifteen-year-old daughter Prospero could be anywhere between thirty and ninety. The Tempest portrays Prospero’s brush with revenge, and can be viewed as an expression of an old man’s indignation, pique and superior power but he learns an important lesson which includes qualities of forgiveness, peace, and humanity.

Prospero is saying goodbye to a life time of magic that, while impressive to others, meant nothing to him while trapped on an Island after being banished and disgraced.

This was Shakespeare's last play and as such, his farewell to the magic of the theatre and the wonder of words. When Prospero says this line, we can imagine Shakespeare speaking through him.