Fry until brown, then flip and fry the other side. Spread the latkes one layer thick on paper towel to drain. If necessary, you can put them in a 250 degree oven, paper towel and all, to stay warm, but they usually don't last that long. Either your oil is not hot enough, or your latkes are too thick. Also try shredding the potato pieces smaller next time.
Latkes fall apart in the oil.
Try using a smaller size shredder. Or add more egg. Or squeeze more liquid out. Or add some more flour or matzah meal. Keep cut but ungrated potatoes covered with cold water.