Well, you have to either keep track of those laws for the states you choose to sell into, or use the free software provided by each state that has opted into the arrangement (the MFA provides an opt-in regime for states, with conditions including the provision of free software, with retailers not liable for any failure to collect and/or remit tax that is due to errors in the software.) But, yeah, you have to pay each taxing "state" (which will probably be less than 50 initially, but more than 50 in the limit case, since "states" under the MFA include the the 50 states, DC, plus other US territories and possessions.)
Or, instead of using the State-provided software and doing it yourself, you can pay a "certified software provider" to do it, and be immunized against liability for any errors that are the fault of that provider.
The current situation in the EU is that if you order a book from the UK to Belgium, you pay Belgian VAT for the transaction. There is also an exception that sellers below a certain volume are exempt from dealing with the cross-border VAT. That is precisely what the new situation in the US will be, if this passes: cross-state purchases will be taxed based on the destination state's sales tax, with sellers below a certain volume exempt.
Despite the dishonest label "internet sales tax" frequently used by the media and opponents, the US measure being discussed "isn't some specific internet tax", either. Its a feederal law that gives permission, with conditions, for states to require remote retailers to collect and remit sales and use tax that are already due on purchases (and which, for remote sales, would otherwise generally have been the responsibility of the purchaser to report and remit). It is neither a new tax, nor is it "internet specific" in any way.