The oracle spoke through a local woman, the Pythia, who was chosen from among the peasants of the area. Documents provided by the Greek historian Plutarch (AD 46-120) describe the Pythia inhaling vapours from a fissure called Adyton. Shortly after she would go into a trance, which enabled her to contact Apollo, the god to whom the oracle was dedicated. Occasionally these trances deepened into delirium and even death, but more normally the woman would utter extremely cryptic and freely-interpretable rhymed answers, which were sometimes conflicting.
According to local legend, these powerful vapours were the breath of Piton, the sacred guardian of the temple. Apollo defeated her and closed her dragon-body in a deep cave under the surface, and the vapours inhaled by the Pythia corresponded to the beast’s breath.
Gatti's article describes the geological formations that made this possible:
The Delphi faults are NNW-SSE trending fractures which cross beneath the oracle site. Where faults join like this they can provide pathways through which ground water and gases can rise to the surface, especially in a tectonically active zone like the Corinth zone (the most important earthquake for this zone was the Corinth earthquake of 373 BC, which probably influenced the reactivation of the fault, as during each seismic event new pathways were created and additional volumes of gas may have been squeezed from the underlying formations).
The gases themselves are likely to include methane, carbon dioxide and
hydrogen sulphide, which can bring about euphoric effects when inhaled.
The photo is from Wikipedia, which also has a video tour of Delphi and lots of other information about this fascinating site.