T. rex went extinct during the K-T mass extinction, about 65 million years ago. This extinction killed the remaining dinosaurs (not just T. rex) and many other animal and plant groups.
This extinction was probably caused by a catastrophic asteroid colliding with Earth. It is thought that a an asteroid 4-9 miles (6-15 km) in diameter hit the Earth off the coast of Mexico. The impact probably penetrated the Earth’s crust, scattering dust and debris into the atmosphere, and causing huge fires, volcanic activity 
, tsunamis
, and severe storms with high winds
and highly acidic rain 
. The impact could have caused chemical changes in the Earth’s atmosphere, increasing concentrations of sulfuric acid, nitric acid, and fluoride compounds. The heat from the impact’s blast wave would have incinerated all the life forms in its path.
The dust and debris thrust into the atmosphere would have blocked most of the sunlight 
for months, and lowered the temperature 
globally.
Those organisms that could not adapt to the temperature and light changes would die out. Since plants’ energy is derived from the sun, they would likely be the first to be affected by changes in climate. Many families of phytoplankton and plants would die out, and the Earth’s oxygen levels may well have dramatically decreased, both on land and in the oceans, suffocating those organisms which were unable to cope with the lower oxygen levels.
Major changes in the food chain would result from all of these these environmental upheavals. The herbivores (plant eaters) who ate those plants that died out would starve soon after these plants died. Then, at the top of the food chain, the carnivores (meat eaters) like T. rex, having lost their prey, would have to eat each other, and eventually die out. Their large carcasses must have provided smaller animals with food for quite a while.
Too big to fail?
Cope’s Rule (named for the American paleontologist Edward Drinker Cope) states that organisms within a population evolve to become more massive over time. Although this increases each individual’s fitness, it leaves the species more susceptible to extinction.
The Lilliput Effect (named by Adam Urbanek, 1993) notes the appearance of small body size in surviving animals after an extinction event. The name Lilliput is from Jonathan Swift’s novel Gulliver’s Travels; in the novel, the Lilliputians were very tiny people.