Transcription and translation base on genes, the chromosomes to be exact. The chromosome is constructed by the DNA. The DNA has the so called nucleobases. A mutation changes those nucleobases inside of the DNA. To make a DNA to an RNA by transcription, it has to be created the complementary bases of RNA. For this step, the transcription has to 'read' the nucleobases of DNA. If the RNA is done, the translation synthesizes the protein on the basis of the given bases of RNA.
Guess, you understand what the consequence of mutation is: by changing the nucleobases of DNA, completely different complementary bases for RNA are created and so a different RNA too (step of transcription). And a different RNA causes a different protein in turn (step of translation).