Garmin Express will automatically launch when a compatible device is connected to a USB port on the computer.

This feature is enabled by default and behaves differently depending on what type of device is connected:

Fitness devices - The Garmin Express window appears automatically when a new device is connected. Once the device has been paired to Express, connecting the device will no longer cause the Garmin Express window to display. However, any new data on the device will be synced automatically in the background.
All other devices - The Garmin Express window always appears whenever a device is connected, even after initial pairing has been performed.

Disabling Garmin Express From Automatically Opening

Open Garmin Express
Select (settings)
Uncheck the box next to Automatically launch Garmin Express when a Garmin device is connected
Select OK

Open Garmin Express
Select Garmin Express in the menu bar
Select Preferences
Uncheck the box next to Automatically launch Garmin Express when a Garmin device is connected
Close Preferences window

Garmin Express will no longer open when a device is plugged in.