Because the hymn stops where it does, the content of the culminating rite is unknown, and the vacuum has been filled repeatedly.
What can be said from the hymn and from scattered later sources is procedural: a procession from Athens to Eleusis along the Sacred Way, a fast, the drinking of the kykeon, entry into the Telesterion, and something seen. Christian writers of late antiquity, hostile witnesses writing centuries later and with an interest in making the rites sound absurd or obscene, supply the most detailed claims, an ear of grain displayed in silence, a sacred marriage, a birth announced. Hippolytus and Clement of Alexandria are the sources for these, and they are not neutral.
The best-known modern hypothesis is that the kykeon contained an ergot-derived psychoactive compound, argued in a 1978 book by Wasson, Hofmann and Ruck. It is an attractive suggestion, it has been influential well outside classics, and it rests on inference about what could have been in barley rather than on any evidence about what was. Classicists have generally not accepted it. It should be described as a hypothesis with distinguished proponents and no direct support.
More recent archaeological and residue work has been claimed as support and has been contested in turn.
The safest statement is the one the hymn makes: something was shown, it was held to change the initiate's relation to death, and the participants kept quiet about it for a millennium. That is an unusual outcome and it is the most interesting fact about the cult.