My mom's toenails look like dead. They are very rough and stop growing. What could be the reason and & treatment?

Fungus nails. Most likely. Treatments vary oral medication, topical s, and laser are available.
Possible fungus. You can not know for sure what it is until a culture of the nail is taken. The nails can be cut and smoothed and then start usinf atopical antifungal to the nails. There are also oral medications and even laser therapy.
Onchomycosis. The appearance of the toenails are likely caused by a fungal infection. This infection can be treated by removing the affted nails, applying a medicated polish or taking anti-fungal medication. It will take several months for the nails to respond to treatment and there is not a 100% cure rate. See your doctor to confirm the diagnosis and treat the infection.