There are probably a hundred ways to do this with string functions. Here's one:
$addr = 'John Doe <john@example.com>';
$explode = explode('<', $addr);
$addr = trim($explode[1], '>');
echo $addr;
Off-topic edit: There is such a domain as "doe.com", and probably a good chance someone there has the username "john". With email harvesters all over the place, it's a good idea to use "example.com" for examples.