We do not know what exactly happens to the X-Men in "Logan". The closest we are told is the radio news report saying that a year before the events of the film, the Westchester event consisted of 600 people paralyzed, and 6 mutants of the group known as the X *Logan Turns off the Radio*....
We know that mutants have become a negative population, having had no natural births in 25 years. Since the idea behind the X-Men is to protect the interests and lives of a threatened minority, it is very difficult to do as the population dwindles. Less mutants means less need for the X-Men to exist. We also know that the X-Men have a sort of nostalgic celebrity status, from the previous decades in the film. Between this, we can only assume that Xavier's school, and his X-Men, have become obsolete and likely are not active. After the Westchester event, Wolverine immediately took Charles and hid him away for his (and everybody else's) own good.
But all this tells us is that some mutants died, maybe part of the X-Men. The rest could still be active, busy or have disbanded.
Keep in mind that "Deadpool" shows us that Xavier's school and the X-Men are active 13 years prior to the "Logan". And also makes Negasonic Teenage Warhead one of the last mutants born before the genetically engineered corn-syrup kicked in. It also means un-natural mutants can continue to be created via Francis' methods ala Deadpool.
Update: They had scripted a flashback scene showing the Westchester Incident but it was cut without filming. It's likely that we will find out when a copy of an earlier draft gets leaked, as that often happens. For now they will keep it intentionally vague.