Yes, it can be really annoying to some people (like me, many of my friends and acquaintances, the Oatmeal, and many others) when people make the mistake of confusing "they're, their, and there" or "your and you're", along with a host of other language issues. However, the following statement fails on its logical basis:
"As a English non native, I don't understand how English natives can make mistakes with there, their, and they're."
As vomitous as the "their, there, and they're" error may be to some like me, the above statement itself is based on the fundamentally flawed premise that increased fluency in a language is directly related language to perfection. For that flawed logic, this statement is itself a facepalm statement of "logic-fail". I mean, am I to expect that native speakers of the kvetcher's language don't regularly make a class of error that learners of selfsame language almost never make? Sorry, but I don't buy it, and here's why:
I believe that the major reason why native speakers are likely to make the "there, their, they're" class of error and non-native speakers aren't is the same reason why native speakers of English write "... I would of done it": they're sounding out the sentence in their head, and (apparently) they have such a horrible mental accent (or an inability to distinguish different words in their mental accent) that they make otherwise simple errors. (Plus, being surrounded by people who don't constantly harp on their mistaken spellings doesn't help with fixing such errors before they become ingrained.)
In addition, there is the problem that - as a language that has a really convoluted history of hybridization of multiple languages (and grammars) and (heavily salient here) highly varied preferences in spelling (and transliteration) over time, it's not surprising that spelling errors are one of the more common errors in written English by native speakers (whereas improper grammar is the more common error in non-native speakers of English). I mean, how many different ways can you pronounce -ough? (Apparently ten ways: enough, cough, drought, though, thought, through, thorough, hiccough, hough, and lough; sometimes, different pronunciations are mixed together, as in Loughborough, and sometimes one word as different meanings, depending on how it's pronounced, as in slough.) How many of the words using ough actually sound like the letters that make the word? (Arguably none.) And yet (and yet!) so many people (children and adults alike) are told to, "Sound out the word." Yeah... in English, it's not as useful a piece of advice as in languages whose orthography better matches its pronunciation.
On the other hand, non-native speakers of English often consciously cogitate creating concordant sentences systematically supporting some sense of grammar that seeks to enshrine a unity between spoken and written forms. Therefore, this type of mistake is less common for non-native speakers of English than errors of fundamental grammar (i.e., those parts of grammar that come so unconsciously fluently to native speakers that they often cannot explain the simple rules of them to non-native speakers beyond the next-to-completely-useless, "It just *sounds* right").
You'd expect a native English speaker to (relatively easily) make the "homophonous" mistake of "they're, their, and there" (or "its and it's" or "pique, peak, and peek" or "site, sight, and cite" or "complement and compliment" or "cavalry and Calvary" or "she and sidhe"), but (almost) never forget to correctly and unconsciously present every single noun as plural, singular or uncountable and either definite or indefinite. With people whose native language doesn't have (or uses different sorts of) concepts of countability, plurality, or definiteness intimately associated with every single noun one utters, making the distinction between, "a cat," and "the cat" (either as a real object in a room or as a conceptual object) really quite difficult.
In other words, each language has its own quirks that create "non-native speaker problems" and "native speaker problems," and I'd wager that - on the whole - the class(es) of problems faced by proficient non-native speakers are more related to grammar than those of native speakers. ... but I guess anyone who worked with non-native English speakers (or have spent time as an adult learning another language) already knew that...
Still, my original bit of pique was that the fundamental logic of the statement is flawed (regardless of the level of personal annoyance I find with people who commit the "there, their, and they're" error), and the statement is thus worthy of it's own, independent, facepalm due to inherent logic-fail.