how can gene mutations happen?

The error-correction isn't perfect. Every so often the wrong base is incorporated. The error rate depends on the enzymes. I don't know the actual numbers but my guess would be from 1 in a 1000 to 1 in 100,000 mis-incorporations.

When only one of the two strands of a double helix has a defect, the other strand can be used as a template to guide the correction of the damaged strand. In order to repair damage to one of the two paired molecules of DNA, there exist a number of excision repair mechanisms that remove the damaged nucleotide and replace it with an undamaged nucleotide complementary to that found in the undamaged DNA strand.

Base excision repair (BER) repairs damage to a single nitrogenous base by deploying enzymes called glycosylases. These enzymes remove a single nitrogenous base to create an apurinic or apyrimidnic site (AP site). Enzymes called AP endonucleases nick the damaged DNA backbone at the AP site. DNA polymerase then removes the damaged region using its 5’ to 3’ exonuclease activity and correctly synthesizes the new strand using the complementary strand as a template.

Nucleotide excision repair (NER) repairs damaged DNA which commonly consists of bulky, helix-distorting damage, such as pyrimidine dimerization caused by UV light. Damaged regions are removed in 12-24 nucleotide-long strands in a three step process which consists of recognition of damage, excision of damaged DNA both upstream and downstream of damage by endonucleases, and resysnthesis of removed DNA region. NER is a highly evolutionarily conserved repair mechanism and is used in nearly all eukaryotic and prokaryotic cells. In prokaryotes, NER is mediated by Uvr proteins. In eukaryotes, many more proteins are involved, although the general strategy is the same.

Mismatch repair systems are present in essentially all cells to correct errors that are not corrected by proofreading. These systems consist of at least two proteins. One detects the mismatch, and the other recruits an endonuclease that cleaves the newly synthesized DNA strand close to the region of damage. In E. coli , the proteins involved are the Mut class proteins. This is followed by removal of damaged region by an exonuclease, resynthesis by DNA polymerase, and nick sealing by DNA ligase.