Either use Wireshark or look at the Curl documentation. On a quick glance, either --verbose or --trace make Curl dump everything it sends or receives.
Again, you will have to learn Curl and HTTP in either case, so now is a good time to start with that.
As I now found out, I already said the same, even more verbose, in Re: CURL and Perl (it rhymes!). So maybe re-reading that thread helps.
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