First make sure you have adb installed. If you don't the ToolsInstallation page has instructions for doing that.
1. First find the latest update file for your device in ReplicantImages, it will look like that:
replicant-6.0-<yourdevice>.zip
For instance for the Samsung Galaxy S2, the update file will be named replicant-6.0-i9100.zip
2. Download the update file on your computer
Reboot to recovery
1. Press the power button
2. Select Reboot
3. Select Recovery
4. Press OK
You can then install the zip with adb. In the recovery:
1. Select Apply from ADB
2. Back to the host computer, load the system zip with sideload:
adb sideload path/to/replicant-6.0-i9300.zip