Graylog stores them in a database and allows you to perform custom searches on them. You can also see in real time how many log messages are being received and processed. The real killer feature in Graylog is streams. A stream in Graylog is a predefined custom search. The server will display to you log messages that only match the stream criteria selected, graphing the results for you.
He mentions other features of the Graylog2 system including customized alerts, messaging via UDP/AMQP and how it uses MongoDb and ElasticSearch. He also introduces Monolog, a PHP tool that lets you log in multiple formats directly to the Graylog2 server. He includes an example of how it works with a sample "LoggingHelper" added as custom exception and error handlers.