If you have a piece saved from when the carpet was installed and that piece has an exact pattern match, you can cut out the burned piece and inlay the new piece. You have to set the new over the burn and cut the old to match the new.

Not easily done otherwise.

How much larger do I cut the new piece? Do I glue the new piece or use double sided tape?? Do you think I can do this myself? Thanks

berber is pretty tough compared to pile type carpet and is stiffer, which actually makes it easier, imo. We used to sew in patches on thicker pile carpet repairs. I would call in a carpet/floor installer for the fix. It will be relatively inexpensive and they will glue tape the seams down with their hot iron, (or sew it), and will make the patch seamless and unnoticeable if they are reputable and experienced. Dependent on where it's at and how much they have to pull up, they have the stretchers/kickers to get it back where it needs to be when layed back down.

Get a carpet installer.

They will have the seaming iron and seaming tape to make as perfect a repair as can be done.

If you have any other seams this is how they were done.