Can some one explain what they did ?

 

They found the voltage across both capacitors

v=[1/(C1+C2)]*integral{i(t) dt}

Then the current through C1 is:

i1(t)=C1*dv/dt=[C1/(C1+C2)]*d/dt[integral{i(t) dt}=[C1/(C1+C2)]*i(t)