In a book on algorithms I read that $n^2 (1+\log n)$ as $n$ approaches infinity is approximated to $n^2 \log n$.
I am not sure if I understand reasoning in this. Is it because $1+\log n$ grows so fast that $\log n$ could substitute it, so that +1 makes no big difference and so is ignored?
Yes, for large $n$ you can consider for all practical purposes that $ n^2 (1+ \log n) \approx n^2 \log n$. Is is like you say, as $n$ increases the +1 becomes more and more insignificant (to quantify how insignificant, you can compute the relative error).