A variant in the coding DNA of a gene changes the bases that are read as codons. If a change to one base turns a codon into one that codes for a different amino acid, the amino acid sequence of the protein changes. A different sequence changes how the protein folds, so it can end up in a different shape.
For an enzyme, a new shape may mean the active site no longer matches the substrate, so the enzyme may work less well, with lower activity, or not at all. This changes the phenotype. In sickle cell disease, one base change in the gene for haemoglobin swaps one amino acid, and this alters the shape of the protein and the red blood cells that contain it.
Not every variant has an effect. Several different codons can code for the same amino acid, so some changes leave the sequence unchanged. Adding or removing a base is usually more serious, because every codon after it shifts and many amino acids change.