Export credentials from your browser

Chrome

Go to chrome://flags
Enable "Password import and export"
Disable "Enable Material Design settings" (With material design, import / export buttons do not show up in the settings)
Go to chrome://settings/passwords
Hit the Export button to get passwords as CSV files

In Firefox 57 addons do not have access to passwords anymore (feature request tracked in this issue), so they also cannot export them. Nor does Firefox provide a built-in option to export them. (feature request tracked in this issue) *

Currently there are the following workarounds:

Import your passwords into another browser (via their importer) and export them there.
Temporarily downgrade/install Firefox ESR, where you can install the add-on mentioned above. Tip: The passwords are saved in logins.json and key3.db. So if you do not want to mess with your main profile, you can create a new profile (just start the ESR version with firefox -p) and copy the files there.
Use third-party tools, such as this one. Of course, you have to unselect "Use master password" in the settings for this, so they are saved unencrypted and other tools can read them.

* Note: You can vote for these issues on Bugzilla, if you want to express your support.