I am sharing a computer with a co-worker. We have two different Trello accounts and we're using the same browser. I need to log out of Trello so I can log into my own account.
To logout, click your profile icon at the top right and select "Log Out"
Note: this will log out of all Trello sessions in the browser, so you and your coworker can't be logged in at the same time unless you are using 2 browsers or are browsing incognito.
You could install another browser (e.g. Firefox or Google Chrome) and use that one to view your Trello-account, your co-worker can use the other browser with his account.
Another option (that doesn't involve installing another browser) might be to have one of you use your browser's regular session, and the other open a "Private Browsing" session in a new window to log into Trello. This way you can both stay logged into Trello at the same time (if that matters).