Determining when and how the first galaxies reionised the intergalactic medium promises to shed light on both the nature of the first objects and the cosmic history of baryons. Towards this goal, quasar absorption lines play a unique role by probing the properties of diffuse gas on galactic and intergalactic scales. In this review, we examine the multiple ways in which absorption lines trace the connection between galaxies and the intergalactic medium near the reionisation epoch. We first describe how the Ly α forest is used to determine the intensity of the ionising ultraviolet background and the global ionising emissivity budget. Critically, these measurements reflect the escaping ionising radiation from all galaxies, including those too faint to detect directly. We then discuss insights from metal absorption lines into reionisation-era galaxies and their surroundings. Current observations suggest a buildup of metals in the circumgalactic environments of galaxies over z ~ 6 to 5, although changes in ionisation will also affect the evolution of metal line properties. A substantial fraction of metal absorbers at these redshifts may trace relatively low-mass galaxies. Finally, we review constraints from the Ly α forest and quasar near zones on the timing of reionisation. Along with other probes of the high-redshift Universe, absorption line data are consistent with a relatively late end to reionisation (5.5 ≲ z ≲ 7); however, the constraints are still fairly week. Significant progress is expected to come through improved analysis techniques, increases in the number of known high-redshift quasars from optical and infrared sky surveys, large gains in sensitivity from next-generation observing facilities, and synergies with other probes of the reionisation era.