Explain the language?
Like many of the characters in many of Shakespeare's plays, Prospero speaks in verse here. He uses iambic pentameter lines (lines that are ten syllables long, with a short or unaccented syllable followed by a longer or accented one), though the cadence is not strict here (some lines are a little longer or shorter).
He speaks to Miranda using "thee," which means that even though it sounds archaic to us, he was speaking informally, as father to daughter.
He is repetitive, making sure points get driven home. He is largely straightforward and literal, though he does use some metaphors and images (about his ex-wife, for example), as well as other figures of speech.
When speaking to Ariel, Prospero is even more direct, and somewhat livelier in tone. He still uses informal address (as would be appropriate for master to servant). His speech is more vivid here—evoking memories, rather than explaining facts.