What can go wrong if we let sigma algebra to admit the union of uncountable union of elements? By definition we only allow the union of countable infinite of elements to be also include the $\sigma$ field, why not uncountable many? Is there a historical view behind this？

The borel sigma algebra, which is used in many cases, contains every single point of the underlying space as an element. If you'd allow uncountable unions, the borel sigma algebra would have to contain any subset of the space. 
However lots of measures cannot be constructed in a way that every set is measurable, a very important example of this is the measure on $\mathbb{R}$ that maps each interval to its length. There is a proof that it cannot be constructed such that every set is measurable.
