It can be repaired, how much depends on how deep the damage goes, generally these type of problems are on boats with plywood core transoms, moisture gets it from the top cap, drain plugs, or bolt holes that mount the motor. Sometimes this damage can be done if you hit a underwater obstruction like stump, rocks, etc but ussually the prop and skeg take the damage and the transom is unscathed.
The cost to fix can be as low as 800 or as high as 3500 , but expect something in the middle to the upper end of the scale. depending on the age of the boat it may not be worth the cost to repair, if you have to pay someone to fix.
You can do it your self if you have mecahnical ability, and willing to get covered in fiberglass dust, and if you do it in your garage etc it too will be covered in that dust, and the chemicals involved are hazerdous to breath, lots of ventilation and a resperator is required. But yo can do it if you choose to do so .. Ive done several for my self.
Cost to do it your self to replace transom if that is the only issue, will be about 300
The only place I know that still does fiberglass work in Chattanooga is Custom Boat works 7717 Hixson Pike 423-842-2000