Complexes is the set of the complex numbers. When used inside Solve it means that you seek complex solutions. Usually it is the default, and you don't need ti specify it.
ComplexExpand is a command to try and rewrite complex numbers into separate real and imaginary parts. For example
In[15]:= ComplexExpand[Exp[Pi I/3]]
Out[15]= 1/2 + (I Sqrt[3])/2
Finally, /. C[1] -> 0 is a replacement rule to replace all occurrences of C[1] with 0.
