Good question - this one hits on some of the subtlety that can make Sentence Correction pretty tough.
With A and B, the modifier "from a one-page writing sample" would apply to the company's claim: "The company claims from a one-page writing sample...". Well, the company isn't making the claim from the writing sample; they're making the claim that they can use the writing sample to achieve a goal. Accordingly, these are incorrect (note: the GMAT will do this fairly often with the phrase "due to", as well, trying to imply that an event was caused by something that couldn't logically be the catalyst).
Answer choice E is incorrect in that the phrase "claims being able to..." is wrong. One quick tip - on the GMAT, the word "being" used as a verb is almost always incorrect (honestly, I have not seen it used correctly on any GMAT questions...but would hate to say "never" in an official capacity).
Answer choice D ("claims to be able to...") is the cleanest way to express that verb construction, as it uses the infinitive form of the verb "to be". When considering whether to use the infinitive (which is often correct on the GMAT), you may want to use your own, similar sentence. You wouldn't say "I decided running the marathon"; instead, you would say "I decided to run the marathon". The authors of GMAT questions are pretty good at creating situations in which the words they use don't jump out as being correct or not, but if you can create a parallel situation in which the correct usage is more striking, that can be a helpful tool.