If you can see fine on the external monitor, then your attached display, or the ribbon cable that connects it, are your problem.
This means the laptop will have to be opened up and the cable tested to see if there are any problems with it first. If there aren't any or is is replaced and still there is a problem, then the display itself will need replacing.
NOTE: If it is under warranty, do not open it up. Doing so would void the warranty.
If the laptop is under warranty, then I would contact the manufacturer about a repair. However, if not under warranty, then I would try a local tech.