I assume this ionising damages DNA leading to mutations, such as cancerous cells.
Correct, ionising radiation creates double strand breaks of DNA which leads to mutations.
How would Gamma-radiation cause this, then, it's simply a high energy photon - how would a photon ionise an atom?
Gamma rays and x-rays (photons) are non-charged particles, they are called 'indirectly ionising radiation'. Electrons (beta) and alpha particles are charged particles meaning they interact directly with matter through Coulomb forces. They quickly loose energy which is why these particles have short ranges.
There are a few ways that photons can interact with matter, each resulting in a release of electrons, which are 'directly ionising radiation', this is where the damage is done.
Photons can interact through:
The photon interacts with the electron cloud of an atom and is 'scattered', loosing some energy and changing direction in the process. The lost energy release an electron from the electron cloud and this electron goes on to deposit energy in the matter.
This is the predominant interaction for high energy photons, its probability also depends on the atomic number of the material.
When the photon has a sufficiently low energy (usually after undergoing several Compton scattering events) it can be absorbed by an atom. To compensate for this an electron is ejected from its shell again and will go on to deposit energy. The photon ceases to exist after this interaction.
There are other by-products of these interactions such as characteristic x-rays or Auger electrons due to the reshuffling of the electrons in various shells.
Photons are massless uncharged particles that have to interact with matter to generate electrons (beta). Electrons are charged particles that deposit energy quickly in matter through Coulomb interactions, this can lead to DNA damage which causes mutations.