While there refers to that place, they're is a contraction for they are. Often people use "there" or "their" when the appropriate usage is "they're".
Comparison chart
 | There | They're |
|---|---|---|
| Used as a contraction | No | Yes - they're is an abbreviation for "they are" |
| Meaning | (in or to) that place | They are |
Definitions
There means:
adverb
- in or at that place (opposed to here): She is there now.
- at that point in an action, speech, etc.: He stopped there for applause.
- in that matter, particular, or respect: His anger was justified there.
- into or to that place; thither: We went there last year.
- (used by way of calling attention to something or someone): There they go.
- in or at that place where you are: Well, hi there.
pronoun
- (used to introduce a sentence or clause in which the verb comes before its subject or has no complement): There is no hope.
- that place: He comes from there, too.
- that point.
noun
- that state or condition: I'll introduce you to her, but you're on your own from there on.
adjective
- (used for emphasis, esp. after a noun modified by a demonstrative adjective): Ask that man there.
interjection
- (used to express satisfaction, relief, encouragement, approval, consolation, etc.): There! It's done.
Video explaining the differences
This video explains the differences between there, their, and they're:
Their 
Their 
Affect 
Advice
Comments: There vs They're