Since the word 'contract' means to shorten or squeeze together, it seems logical that a contraction is two words made into one by putting an apostrophe where letters have been omitted. Words like can't (can + not), don't (do + not), and I've (I + have) are all contractions.
Usage of Contractions in Sentences
They're is a contraction for they are.
Let's rewrite the given sentence using contractions: They are happy to see me.
Its contracted form would be: They're happy to see me.