Why is identity element required for groups? I would like to know the necessity for having an identity element for every group.
I know the meaning of an identity element.
Most definitions of general objects in mathematics are a consequence of abstracting the properties of certain key examples that the object was originally developed to study so it could be extended to use in other fields. For example, the theory of measure and integration developed from precursor theories that were generalizations of ideas that ultimately had their roots in area and volume formulas that arose in construction problems in Ancient Greece.  

Groups originally arose in the study of the symmetry properties of geometric objects under the classical isometry and similarity transformations of Euclidean geometry. It was Felix Klein who recognized that the set of isometries in Euclidean space-rotation,reflection and identity-formed a group under composition of functions since the operation is associative,yields a unique identity function and an inverse for each mapping which yields the identity map when composed with the mapping. For example, a reflection of an object through a mirror plane is it's own inverse since applying the map twice successively leaves the object unchanged and is therefore the identity mapping.  



The generalization of the group of transformations of necessity gave an identity element in the definition of group when Leopold Kronecker proposed the abstract definition at the end of the 19th century.
By requiring having a unit $e\in G$ you achieve to things: 


Avoiding dealing with the concept of empty sets as the base of the Group.
Avoiding dealing with empty operations (as an operation is a subset of $(G\times G)\times G)$. 


Since the empty set and empty operation have all the other axioms of groups there could be an empty Group. The empty group will be contradicting almost every theorem.

Also, requiring a unit element gives a meaning to having an inverse element and to solving equations.

From the comments below I have missed one main thing. A Group is a structure that exists in nature a lot.