I assume, unlike the commenters, that you have no doubt that "thus [verb]ing" is grammatically correct, but are concerned that "providing" does not make sense, that is, "providing" is not used properly, from a semantic standpoint.
On this, I agree with you whole-heartedly.
How can obtain[ing] thus provide anything? Answer: it can't.
To whom is the reader providing a background? Answer: To nobody. He is obtaining that as well (but in WA2's rendering, apparently the reader is providing background to himself)
As commenters have shown, the only way to make sense of this is to add or change words, or to imagine "missing" words.
I believe you can salvage the provide verb, but it needs to have as a subject a better "feel" rather than the reader.
To do this, simply add "which" amd change "providing" to "provides:
- The reader obtains a better "feel" for the way Dillard interprets the natural world, which provides background and a foundation for the rest of the novel.