What does $\left|\bigcup\limits_{n=0}^{10}\{n\}\right|$ suppose to be?

It is a union:

$$\bigcup_{n=0}^{10} \{n\} = \{0\}\cup \{1\}\cup\ldots\cup\{10\} = \{0,1,2,\ldots,10\}$$

But then, you take the cardinality of the resulting set: $$ \left\lvert \bigcup_{n=0}^{10} \{n\} \right\rvert = \left\lvert \{0,1,2,\ldots,10\} \right\rvert = 11 $$ and you get $11$, as the set contains $11$ elements.

It's the cardinality of the union of all the singlets containing $n$.

So it's the cardinality of the set ${\{0,1,..,10}\}$ which is $11$.