Why is $\sum_{i=1}^n O(i)$ not the same as $O(1)+O(2)+\dots+O(n)$?
The difference is IMHO well explained in that book, the part you are referring to says


  The number of anonymous functions in an expression is understood to be equal to the number of times the asymptotic notation appears.


So this $$ \sum_{i=1}^n O(i) $$

means a placeholder for $$ \sum_{i=1}^n f(i), $$

where $f(i)$ is an anonymous function from the set $O(i)$ &mdash; but it is always the same function in all terms of the sum. However, in $O(1) + O(2) + \dots + O(n)$, each term represents a placeholder for a (potentially) different anonymous function $f_i$ from the set $O(i)$. Note the book also says, the latter has "no clear interpretation", so it is quite meaningless.