How can a mutation in a sequence of DNA affect the function of a protein?

Firstly, it is important to recap how DNA codes for synthesis of proteins. mRNA is formed from complementary base pairing to the template strand of DNA in a process called transcription. Every 3 bases within a gene codes for one codon in the mRNA sequence. This mRNA then leaves the nucleus and binds to a ribosome in they cytoplasm. Here, tRNA molecules with complementary anticodons pair up to their respective mRNA codons. tRNA molecules have amino acids attached to them and hence the mRNA sequence determines the amino acid sequence (the primary structure of the protein). Peptide bonds then form between these amino acids. The primary structure then folds via hydrogen bonds into an alpha helix or beta pleated sheet (secondary structure). The secondary structure then folds via covalent and ionic bonds between amino acid side groups into a tertiary 3D structure.
A mutation changes the sequence of bases in DNA and hence the triplet code. It therefore changes the sequence of amino acids in the protein's primary structure. This changes the side groups that are available to form covalent or ionic bonds to form the proteins specific tertiary structure. Hence, the protein will fold abnormally leading to a change in the structure of the protein's active site (if it is an enzyme) - affecting its binding of substrates and function.