1) Links to any blacklisted site will get removed from US search engines, directories, and maybe even blogs through lawsuits. Google, Twitter, maybe even HackerNews.
2) Your non-US site can get blocked to all US users (ouch) and lose most of its search / social traffic (see #1)
3) Non-US sites can get cut off from US-based payment processors and advertisers. This will push sites you use into bankruptcy, and discourage others from starting.
4) Hollywood and (I'm ashamed to say it) US trade representatives and embassies will be busy convincing other countries to pass similar laws. If this law passes, that process goes better.
5) Countries that want cover for their own internet censorship regimes will be able to say "The US does it". Russian politics in particular loves these (sometimes false, sometimes not) equivalencies as excuses for anti-democratic measures.
As for how you can prevent it, if your run a site or product with content in English, your site has American visitors, potentially lots of them. Same goes for just spreading the word on social networks.
Participate in American Censorship Day and direct them to call their reps!