Understanding JOINs and why the syntax works the way it does
The answer is in the parentheses:

SET price=(SELECT price FROM new WHERE new.id=original.id) 


This is a scalar subquery.  It only ever returns one value (or no value, in which case it effectively returns NULL).  As @ypercube mentioned in comments, if there were ever more than one matching id in new then this expression would throw an error, since a scalar subquery can't and won't deal with more than one possible value.  

This is also (by at least some definitions) a correlated subquery, containing a reference to a table (original) that is not mentioned in the FROM clause of the subquery.

The subquery is, essentially, executed once for each row in original in order to find the needed value in new, and that's how the rows don't get mixed up -- this expression is evaluated for each row in the original table.  

At least, conceptually, that's how it happens.  The optimizer is free to decide --  within the limits of its design -- if there's a better way to perform your query than the way you've written it, as long as the chosen approach will still provide the exact same result (with the exception of the ordering of rows, which is by definition, undefined, unless you explicitly ORDER BY in a SELECT statement).  MySQL 5.6 brought some changes in subquery optimization that were largely improvements.

Although sometimes they are absolutely essential, subqueries can be a red flag that a query's logic could be improved, to make things easier on the optimizer and get the work done faster.  This is one of those cases.

A better, arguably clearer, and perhaps significantly better-performing way to write the example query would be this:

UPDATE original o
  JOIN new n ON n.id = o.id
   SET o.price = n.price;


There's no need to include WHERE n.id = o.id because the join will not only join the rows on that criterion, it will also exclude all rows that can't be joined.  The caveat here is that if id in the new table isn't unique, for a different reason than the error the original query would throw.  In this case, the result isn't deterministic, because you're not able to tell the server which of the more-than-one potentially matching row, so it will pick one, and you can't choose which one.  But if id is unique in the new table, there's no problem.

MySQL has historically been not the best at handling WHERE ... IN ( ... ) in some cases, and this rewrite also removes the need for that construct.