There goes back to Proto-Germanic *þar; it should have become þær in Old English, but the vowel was lengthened, giving þǣr (the same development is seen in where < *hwar).

They're is they + are, of course, and their is the possessive of they (cf. you : your). This pronoun is somewhat interesting, as it was actually borrowed into (northern Old) English from Old Norse. The original third-person plural pronoun in Old English was hīe, which is the paradigmatic plural of his < OE hē and it < OE hit. They comes from the Old Norse 3rd pl pronoun, þeir.

Different Germanic languages used different Germanic pronouns as their basic personal pronoun for the third person: English used descendants of *hiz 'this', German eventually settled on descendants of *iz 'he', and Old Norse went with *sa 'that' (other case-forms of *sa began with **þ-*, don't worry about that). It's actually much more complicated than that makes it seem, but let's not get into that.

Anyway, it's fairly common knowledge that England experienced a lot of Norse raiding towards the end of the first millennium CE. Many Norsemen came to England and then stayed; this is why there are so many words in English from Old Norse. Anyway, one such word that made it across was they. It's common for languages in contact to borrow words from each other, but it's exceedingly rare for a language to borrow 'core' vocabulary items, like pronouns and words for body parts &c. This shows that there was intensive contact and a high amount of bilingualism in the Danelaw (the area where the Norse were heavily settled). Anyway, they spread from there and eventually squeezed out the descendants of hīe (this was surely aided by the fact that the forms of hē and hīe became very close in pronunciation). The oblique form hem probably survives in the colloquial truncation 'em.

Now (whew), how did they come to be pronounced identically? Not too hard. In English, there have been a lot of mergers of vowels before r. We can compare the similar trio Mary, marry, and merry; in many dialects of English these are homophonous (and in some two of them are). It's a slightly different case for our three words, but it's the same principle. To put it very crudely, the consonant r affects the acoustic properties of a preceding vowel. This can make it more difficult to distinguish relatively close vowels in this environment, and learners may undergo a merger. There are many of these.

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It has to do with the way people think of language. Many people think with mental voices, and words to them are "sounds" or voices in their heads rather than abstract units of meaning. Thus, many people often mix up homophones, their/there/they're being one of the most common examples, but the same is true of affect/effect, than/then, your/you're, and less common ones including principle/principal and emigrate/immigrate. People who do not read or write very often will have less exposure to the proper usage, and thus be more accustomed to hearing and speaking the words, where no distinction is made.

This is the same reason why so many people don't know the correct ways to write common phrases or words:

• It doesn't phase me - faze

• Should/could/would of - should've/could've/would've

• Could care less - couldn't care less

• Chester drawers - chest of drawers

• For all intensive purposes - for all intents and purposes

Also, this doesn't directly address your question, but I just wanted to say that oftentimes, native speakers will not be as precise with their language as people who have it as their second language. Native speakers usually pick the language up through osmosis, as opposed to people who actually "learn" the language formally at school, with all the grammatical rules and proper structures.