The asterisk, like many other symbols in C, have multiple meanings depending on context. In a variable declaration it means the variable is a pointer, used in an expression like this it dereferences a pointer, that is it gets the value of what the pointer points to.
Then you have the postfix increment operator ++ which returns the value of the expression and then increases it, in this case it returns the pointer and then increases the pointer.
So what e.g. *src++ does is dereerence the pointer src to get its value, then increase the pointer (so it points to next location in memory).
As for the whole expression *dest++ = *src++ it simply copies the value pointed at by src to the value pointed at by dest then increase respective pointer. In short, it copies from src to dest. You will most likely see this in a loop.