Re: And how exactly
@Neil Barnes: "genuinely interested in the legalities here"
IANAL, you need to consult a real one (L, I mean), but I can wave my hands as vigorously as the next guy.
I assume, as a practical matter, that Snowden gets any royalties through the publisher. I assume the publisher can be leaned upon if they are American or if they ever want any product of theirs to reach American audiences. Even if Snowden's book is not distributed in the US, but other products from the same publisher are, this means the publisher has some legal presence in the US, US banks are involved, etc., and therefore a US court order may well apply ("you, Mr. Publisher, owe the US Government the fees due to an author of yours, Mr. Snowden, $5M in total, therefore your US bank account(s) and other assets are frozen until such sum is paid in full", etc., etc. - whatever the procedure is).
Note that the US Govt has not asked to restrict the publication, and it only claims Snowden's fees, not the publisher's profits. That probably increases the chances that the publisher will co-operate smoothly, and strikes me as smart: cut off Snowden's income, but don't try to suppress information that is out there already - kinda similar to invalidating his passport as a means to travel, but not his citizenship.
The citizenship part means, among other things, that Snowden has to declare his income to the US Govt and to pay his taxes, even on income earned outside of the US. If he doesn't he becomes guilty of breaking devil only knows how many other laws.
treaties which allow some criminal cases to be pursued outside the borders of the USA
This is a civil suit, completely separate from the criminal case against Snowden (check the link to the DoJ PR in the article), and maybe there is no need to pursue anything abroad, cf. above.