directs people to route their mail to computer bigsite (presumably a well-known location accessible to everybody) and from there through the computer foovax to the account of user me on barbox.
Before autorouting mailers became commonplace, people often published compound bang addresses using the convention (see glob) to give paths from *several* big computers, in the hope that one's correspondent might be able to get mail to one of them reliably. e.g.
...!seismo, ut-sally, ihnp4!rice!beta!gamma!me
Bang paths of 8 to 10 hops were not uncommon in 1981. Late-night dial-up UUCP links would cause week-long transmission times. Bang paths were often selected by both transmission time and reliability, as messages would often get lost.