Why should I use the binomial theorem to solve $(1+i)^8$?

You did use the Binomial theorem, not for $8$-th degree, but for $2$nd. The hint does not insist on using it directly to the $8$-th degree, does it?

As an alternative: $$(1+i)^8=[(1+i)^4]^2=\require{cancel}\left[{4\choose 0}+\cancel{{4\choose 1}i}-{4\choose 2}-\cancel{{4\choose 3}i}+{4\choose 4}\right]^2=16.$$