What is monoid homomorphism exactly?

Question

I\'ve read about monoid homomorphism from Monoid Morphisms, Products, and Coproducts and could not understand 100%.

The author says (emphasis original):

Answer 1

Colloquially a homomorphism is a function that preserves structure. In the example of the length function the preserved structure is the sum of the lengths of two strings being equal to the length of the concatenation of the same strings. Since both strings and integers can be regarded as monoids (when equipped with an identity and an associative binary operation obeying the monoid laws) length is called a monoid homomorphism.

See also the other answers for a more technical explanation.