Your begonia looks like it has been attacked by a fungal infection and the fungal growth is the brown material on top of the soil. Damp, cool weather and poor air circulation encourages molds, mildews and fungi on plants especially in the winter. Sorry to say the best thing to do is to toss out the plant and soil and wash the pot well so as not to spread the disease. Use a potting mix with better drainage (or add one third sand or perlite to your regular mix) and be careful not to water until the top one inch of soil is dry. Spring is coming and with warmer nights the fungus among us will not be so quick to attack. Keep Growing, Marianne Binetti