Why Should We Normalise Relational Database?

A database which is not normalized would have relations of the same data. This means that to update the database you would need to update the same data in several different places. This would tend to slow down the database access and make it larger than it needs to be.

That was a very common answer with a practical point of view eliminate redundancy in other words.

Other more technical answer is that Relational Algebra is equivalent to Predicate Calculus (with closed world assumption and other details not to talk about them now).

Normalization is needed because it is equivalent to quantification in Predicate Calculus.
It is not the same to say: For all boys thee exists one girl they are married with  than one girl is married with all boys (polyandry)
quantifiers for all, exist, exist just one, are very subtle to use. If done careless many mistakes may occur, also with its counterpart in relational algebra, normalization.

In synthesis:  If you do not normalize your database will not represent the world you want to.