In your sentence, "newspaper" is defined naturally by "this" but in the original sentence in Arabic, "newspaper" is not defined. To get a bit more into the logic: The sentence in Arabic above, has the "big newspaper" altogether as a predicative (i.e. the information the listener needs to know). While in your sentence, "this newspaper" is the subject, and the predicative (the information the listener needs to know) is "big" (also here it is called a predicative adjective, because it is not attached to the noun directly to describe it).
You sentence would be rather translated as: هذه الصحيفة كبيرة (notice how صحيفة became definite with AL)
شكرا! Makes perfect sense now. So if I add an article to the adjective (making it an adjectival phrase) what happens to the sentence with the demonstrative pronoun (هذهِ at the beginning)? Thank you in advance.
Well, not sure how do you mean with adding an article to the adjective, but in Arabic, the attributive adjective (which comes attached to the noun) is a "follower" to the noun. It follows the noun in gender, number and definition. So, if the noun is defined, the adjective is defined as well; If the noun is dual or more than 2, the adjective is so as well, and of course the same with being masculine or feminine.
If you mean something like: This is the big newspaper, that would be: هذه هي الصحيفة الكبيرة - the pronoun (هي) is somewhat a form of emphasis and you can imagine it (only in this instance here) as a replacement for the (to be) verb in English.