Why is an uncountable union of null sets not necessarily a null set?

A null set is a set of zero measure; more informally, it is a set of points that has no area. For example, all countable sets— including the rationals $\mathbb{Q}$, the natural numbers $\mathbb{N}$, the empty set $\varnothing$, and singleton sets like $\{0\}$—are null sets in $\mathbb{R}$ [under the Lebesgue measure].

Any countable union of null sets is still a null set, but an uncountable union might not be. For example:

$$[-1,1] = \bigcup_{x\in[-1,1]} \{x\}$$

The inverval on the left is not a null set; it has length 2. However, we can write it as an uncountable union of singleton sets that are each measure 0. Hence an uncountable union of null sets can yield a non-null set.

Take any set $A$ which is not a null set. Then $A=\bigcup_{a\in A}\{a\}$. Now, note that each $\{a\}$ is a null set.