ROM means Read Only Memory, you can read from this memory only.
Now u have Two $8$ bit multipliers, total input combinations possible are $2^8 \times 2^8 = 2^{16}$
In the ROM, u have to store output for any possible input, output can be 16 bit long, And there are total $2^{16}$ inputs possible, therefore size of ROM = $2^{16} \times 16$
Now you already computed output for any possible input, your job is to just read the output corresponding to ur current input.
https://i.stack.imgur.com/TNLtQ.png