The exhaust from the fireplace insert has to go somewhere, and the most convenient place for it to go is up the existing chimney. In the simplest kind of insert installation, a short connector pipe runs from the insert's exhaust outlet through the existing fireplace damper and into the chimney flue. Although this kind of installation is initially easier than installing a full chimney liner, it can be costlier in the long run because you must remove the insert to clean the chimney.
An alternative to a short connector pipe is a stainless steel chimney liner that runs all the way from the liner's outlet to the top of the chimney. A full liner eliminates some of the problems associated with short connectors, including the tendency of connectors to allow smoke from the insert to flow around the connector and come back into the room through the damper. An insert with a chimney liner is permanently installed, and you can clean the liner without removing the insert.
Whether the insert is installed with a short connector or a full liner, the space around the insert's outlet pipe where it passes through the damper should be sealed so that cold air from the chimney can't enter the space around the insert. This seal can be made simply by stuffing the space around the outlet with fireproof mineral wool, but a sheet metal block-off plate that covers the opening under the damper creates a more effective and permanent seal; the insert's outlet pipe passes through a hole in the middle of this plate.
When the size of the insert is smaller than the opening of the existing fireplace, there will be gap around the insert. A face plate covers the gap and provides a decorative finish to the insert installation. In the case of a gas-burning insert, the face plate may also include the controls for the insert's ignition, flame adjustment and exhaust fan, if it has one.