Firstly, make sure you have the right size mantle to replace the existing one. They come in various sizes with the most common being S1, S2, D1, D2. With the letter indicating whether it's a single or double tie, and the number indicating the size of the mantle.
Remove the old, broken mantle from the lantern. It should fall apart and easily be blown off into a rubbish bag.
Once the old mantle is removed, tie (or clip) the mantle to the lantern. Depending on what style lantern you have the mantle will either tie to the top and hang, or slide over the burner head and tie, top and bottom.
Mantles are cloth and when new they are soft. It is necessary to burn the mantle before first use for it to work effectively. To do this, simply light it with a match and wait for the mantle to burn itself out. When the mantles are burned for the first time, the cloth fibres burn away and leave a much smaller size ball of ash which when heated produces a bright incandescent light.
Also be aware that once the mantle has been burned into ash, it is very fragile and can fall apart if touched, so it's best to be careful when lighting the lantern. It may be handy to place the lantern in a carry bag for transport to avoid the mantle breaking whilst travelling.
Once the mantle is burned, you are ready to light your lantern.