BUSD.e on Avalanche is a token representing BUSD on Avalanche that was bridged from the Ethereum blockchain via the Avalanche Bridge.
For example, when sending BUSD over the bridge to the Avalanche Network, it will appear as BUSD.e once the transaction is completed.
If you don't immediately see the BUSD.e after a successful transaction, you may need to add the token to the wallet you are using so it shows your balance of BUSD.e.
Note that to send BUSD.e back, is the same process but in reverse, resulting in BUSD.e turning into BUSD.
For a more in-depth read on how the Avalanche Bridge works, please read the following articles.