Return results of cURL shell script direct to AS?

You can try:

set myFile to POSIX path of (path to desktop as text) & "curled.txt"
do shell script "curl 'https://discussions.apple.com/thread/4656790?tstart=0' -o " & quoted form of myFile
set myData to do shell script "cat " & quoted form of myFile

or:

set myData to do shell script "curl 'https://discussions.apple.com/thread/4656790?tstart=0'"

See Apple Technical Note: 2065

The resuts of the shell script can be directly put into an AppleScript variable, the only thing you need to watch out for is escaping any double quotes or backslashes - your command doesn't download or put anything into a file, though.

set theAnswer to (do shell script "curl --data 'browserRequest=true&lat1=43%C2%B043%2738.11%22&lat1Hemisphere=N&***1=69%C2%B04 9%2752.85%22&***1Hemisphere=W&startYear=2013&startMonth=1&startDay=1&resultForma t=csv' http://www.ngdc.noaa.gov/geomag-web/calculators/calculateDeclination")

The results are HTML, so I am guessing you have something that is going to parse that.

The solution is simple - your original shell example uses quoted strings, and you can't just copy/paste that into an AppleScript because AppleScript also uses quotes to delineate strings.

The answer lies in escaping the quotes, so that AppleScript knows to pass the actual quote symbol to the shell (and therefore onto curl) rather than have AppleScript interpret it itself.

do shell script "/usr/bin/curl --data \"browserRequest=true&lat1=43%C2%B043%2738.11%22&lat1Hemisphere=N&***1=69%C2%B0 4 9%2752.85%22&***1Hemisphere=W&startYear=2013&startMonth=1&startDay=1&resultForm at=csv\" http://www.ngdc.noaa.gov/geomag-web/calculators/calculateDeclination"

Note how the quotes surrounding your data are escaped.

(Note also that the above command generates a 500 error off the server because the parameter data is invalid, but I assume you can fix that)