The Strawberry Begonia plant, or Strawberry Geranium as it is sometimes called, is neither a begonia nor a geranium, but rather a member of the Saxifrage family. It does have bluish-green, fuzzy leaves like those of a begonia and it does spread by sending out “stolen” or “runners” like a strawberry plant; but a Strawberry Begonia is really an evergreen native to Asia and a close relative of the Piggyback plant. A Strawberry Begonia produces a lovely, white flower with pointed petals and a bright, yellow center. Like a strawberry plant, new babies develop at the ends of each of the “stolen.” Indoors or on your porch a Strawberry Begonia makes a beautiful hanging plant. Outdoors it can also be an excellent ground cover in your garden.
Your Strawberry Begonia is probably getting some direct sun and warm temperatures during the day. The sun put the holes in the leaves and the high temperatures caused the plant to wither. Move your Strawberry Begonia back inside to a cool low light area, continue to water until it starts to recover; then move it back into bright indirect light.
Strawberry Begonias have many names. Saxifraga stolonifera is its scientific name and the nickname “mother of thousands,” came from its many leaves. A few of its other names are Strawberry Geranium, Beefsteak Geranium, Creeping Saxifrage, Creeping Rockfoil, Strawberry Saxifrage, Magic Carpet, and Aaron’s Beard.
There are a couple of ways to plant these baby Strawberry Begonias. In either case, wait until the baby is a decent size and has several of its own leaves before propagating it. The first method is to cut off the baby with a section of its stolen (runner) and plant it in a small pot. The second way is to place a small pot of soil next to the mother plant and pin the baby down using a piece of wire or an open paper clip leaving the new plant attached to the original plant until roots develop. I always recommend using soil that is very similar to the soil used in the mature plant.
The round soft velvety leaves of a Strawberry Begonia can be solid green or can be green with prominent white veins, which is the variety you seem to have. There is also a variety that has white tips.