Is a definite article (the) necessary for "most"?

Both are correct. Without "the" is preferable in my opinion under the rule of omit needless words. There is no why. Grammar is arbitrary. It's just the way we do it.

"...fascinates me most..." seems to me preferable. The word "the" doesn't add any value. Also, you could rephrase the sentence as "Of all the things I've learned about the textile industry, I am most fascinated by the fact that it drives..." or Of all the things I've learned about the textile industry, most fascinating to me is the fact that it drives..." Both of these formulations, to my ear, do not call for "the" before "most". So I think it can well be omitted in the original sentence.

I think both are correct, but I vote to leave out the “the”.

You mentioned this is Formal Writing you're adding this sentence to, so I would use the in front of the word most. Worth mentioning however, non preferably, the word most is able to stand on its own in this sentence. In addition, I would omit the needless word "That", and replace it with the more appropriate comma after the word "Is". So it reads; "What fascinates me the most about the textile industry is, it drives the economy of many third world countries.". "The most" is also preferred as it is the superlative of the action verb "drives".

The second sentence without the article "the" is correct. I feel it is so as to flow of the sentence.

Both are acceptable. It is purely a matter of preference, and no editor should complain about either version.
English grammar developed in much the same way: A particular preference becomes the most used, and then becomes the standard rule. Some American usage is being absorbed into UK usage through TV programmes (or programs if you are American)

The second sentence is correct because "What facinates me most. . .
"me" is the direct object and most is the adverb modifying to what degree the person is "fascinated."

Convenience of packaged goods

From what I've noticed, it's pretty much everywhere

Both sentences are grammatically correct, but they convey slightly different nuances.

The first sentence, "What fascinates me the most about the textile industry is that it drives the economy of many third world countries," uses "the most," which emphasizes a specific aspect or element that is of greatest interest to the speaker. In this case, it suggests that among all the fascinating aspects of the textile industry, the economy-driving aspect is the most captivating to the speaker.

The second sentence, "What fascinates me most about the textile industry is that it drives the economy of many third world countries," omits "the" before "most." This version suggests a more general fascination with the textile industry overall, without specifically highlighting one particular aspect as the most fascinating.

Both versions are acceptable depending on the intended emphasis or tone of your writing. If you want to emphasize a specific aspect as the most fascinating, use "the most." If you want to convey a more general fascination without specific emphasis, use "most" without "the."