How can I quickly delete multiple messages at once in Discord?

Add Mee6 bot to your Discord server. One of its moderator options is a !Clear command.

Usage:
!clear @someone Delete the messages of @someone in the last 100 messages.
!clear xx Delete the xx last messages. Maximum of 1000. Higher numbers may take a while and can crash your discord on mobile devices.

Alternatively, if adding bots isn't an option, you can make manual deletion a bit easier by holding down Shift. This will skip the confirmation box that usually displays when deleting messages.

If you want to delete all the messages from the channel, Clone then delete the channel.

If you need to delete more than 1000 messages that are less than 2 weeks old, my discord bot PvPCraft can help you, it's harder to use though so for less than 1000 I would go with mee6.