Essentially it does this:

# let's assume some values for the sake of discussion
$a = 'foo'; $b = 'bar';

$a = $a ^ $b;
# $a now contains both variables, $a and $b logical or'ed
# together, so at this point a=foo^bar and b=bar

$b = $a ^ $b;
# logically or-ing them again (since b still contains bar)
# returns the original value of a, so now a=foo^bar
# and b=foo

$a = $a ^ $b;
# then we logical-or the combined value with b, which now
# contains our original a value. this returns the
# original b value, so now a=bar and b=foo
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It's one of those things that you can look at a dozen times, and suddenly your brain just goes 'ah ha!'.

Update: Of course read 'exclusive or (or xor)' everywhere that I said 'logical or', it must be bedtime.

$a = ord('A');  # 01000001
$b = ord('B');  # 01000010
$a = $a ^ $b;   # 01000001 XOR 01000010 = 00000011
$b = $a ^ $b;   # 00000011 XOR 01000010 = 01000001
$a = $a ^ $b;   # 00000011 XOR 01000001 = 01000010
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Or, if you think in terms of truth tables:

As you can see, original $a is the same as new $b and original $b is the same as new $a.