Some historical shells implemented a very simple parser that could get confused by things like [ -n = "" ] where the first operand to = looks like an operator, and would parse this as [ -n = ] or cause a syntax error. In [ "x$1" = x"" ], the x prefix ensures that x"$1" cannot possibly look like an operator, and so the only way the shell can parse this test is by treating = as a binary operator.
All modern shells, and even most older shells still in operation, follow the POSIX rules which mandate that all test expressions of up to 4 words be parsed correctly. So [ -z "$1" ] is a proper way of testing if $1 is empty, and [ "$x" = "$y" ] is a proper way to test the equality of two variables.
Even some current shells can get confused with longer expressions, and a few expressions are actually ambiguous, so avoid using the -a and -o operators to construct longer boolean tests, and instead use separate calls to [ and the shell's own && and || boolean operators.