Adding parentheses to the second statement to show precedence, we get:

( 1 ? ( $x=1 ) : $x ) = 0;
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As the 2nd and 3rd arguments are lvalues, you can assign to the ternary operator. As the first argument (1) is TRUE, the assignment becomes:

( $x=1 ) = 0;
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Which effectively boils down to:

$ perl -Mstrict -Mwarnings -E 'my $x; ($x = 1) = 0; say $x;'
0
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The link I gave above (perlop manpage - under Conditional Operator) has a fuller description.