Okay, just a heads-up: I have not read that book that you are using (I Sparknoted it) so you will have to compare what you know about the book to what I am assuming about it to ensure accuracy.
For the first blank (make sure you paraphrase a little bit, because if your teacher puts your paper through a plagiarism checker, the exact words might lead her to this post and you might lose credit): This piece of evidence is a small-scale example of how such an event could stimulate actions being made toward the British colonization of what is now the United States. It contains direct information on why the present-day United States was colonized (fertile soil, inviting, good climate), and establishes that Raleigh claimed this region. It also supports my initial claim by stating that Raleigh named part of the region after Queen Elizabeth, thus implying that the persecution in Elizabethan England inspired his colonization of this land.
For the second blank, I would use this article that I found and paraphrase this quote from it: "Raleigh’s initial forays into the colonisation of America were funded by private investors. He sponsored a number of attempts to establish an English colony at Roanoke Island, which he named 'Virginia' in honour of Queen Elizabeth ‘the Virgin Queen’ in 1585." Start the phrase with "According to an article by the Royal Museums Greenwich..." Also, if you are American, remember to change the spellings of "colonisation" and "honour" to the American-English spellings (z instead of s; drop the u)
Third blank: https://www.rmg.co.uk/discover/explore/queen-elizabeth-i-colonising-america (But check if your teacher/professor needs to you cite it in MLA or something rather than just the link).
Fourth blank: In regard to how this supports your claim, I would write something similar to how the first piece of evidence supported it. Of course, you can always read the rest of that article as it definitely has some other options for evidence, but this might be the easiest one. Just tie whatever you find back to your first claim (and don't overthink it too much).
I really hope that this helps some. Best of luck with writing the full draft!