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can anyone please explain how multiplication of generating function can be done as given in theorem

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It's simple mathematics .. 

For sake of simplicity , 

Let  f(x)  =  a0 + a1 x +  a2 x2 + .....+ ak-1 xk-1  + ak xk  .......

     g(x)  =  b0  + b1 x  +  b3 x2 +.....+ bk-1 xk-1  +  bk xk ......

So if we consider the coefficient of  xk  in f(x) . g(x)  ,

Then we have   ( a0.bk  +  a1.bk-1 + ................ + ak.b0 )  ..............(1) 

So general term of above summation (as shown by (1)) can be wriiten as  :   aj . bk-j    where j varies from 0 to k..

So it can be written as : Σ aj . bk-j    where j varies from 0 to k.

Likewise we do for every xk  where k starts from 0 and goes to infinity..

Hence  f(x) . g(x) will contain all the terms of 'x' having exponent from 0 to infinity

which is nothing but    =  Σ ( Σ aj. bk-j ) xk   where the limits of inner summation is from 0 to k as explained above and limits of outer summation is 0 to infinity..

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