Around 75% of all plant and mammal species were made extinct by the "K-Pg extinction event" that happened 66 million years ago. Very few large animal species (Wikipedia gives a maximum mass of 25kg) survived - some crocodiles and turtles being the exceptions.

While we tend to think of extinction events as being sudden and immediate (which they are on evolutionary or geological scales), the K-Pg extinction event is thought to have lasted anywhere from a few years to 10,000 years, before things stabilised (similar to the mass-extinction event the Earth is currently going through; hard to notice on a day-to-day basis or year-to-year, but definitely noticeable over thousands of years). A meteor didn't just crash and kill and the dinosaurs - a meteor crashed and caused massive, long-term changes to global climates and ecosystems that lasted thousands of years, and most large animal species didn't survive.

In terms of what happened, the main thing was anywhere up to 2 years of global darkness caused by all the dust and other matter thrown up into the sky by the initial crash. This would have killed off a huge chunk of plant life (which requires photosynthesis to live), as well as likely making things colder. Without plants, animals that lived off plants would gradually die off, and without them animals that lived off other animals would slowly die off - including the big super-predators and super-herbivores like the larger dinosaurs (who would have needed to eat a lot of stuff). Many individuals would survive for a while, maybe some generations, but over time the species wouldn't, as not enough would be around to maintain a stable population.

On top of that, you have a huge chunk of weird minerals and materials being thrown up into the atmosphere, to settle across the globe, some of which would be toxic to some animals and plants. There's also a good chance that some of these minerals settled into the oceans, significantly increasing the acidity of the oceans, making them uninhabitable for a lot of plants and animals, and generally messing up the world's water supply.

Things that survived were generally things with the ability to survive long periods without food (like crocodiles), things that lived off detritus (all the dead plant and animal matter caused by the destruction) like worms, snails and insects, and the smaller animals that lived off them (like the surviving birds and mammals).

And as the dust settled (both figuratively and literally) there was an explosion of diversity in life - with so many evolutionary niches now opened up, mammals in particular flourished, evolving in all sorts of new ways (including getting larger and smarter).

Some dinosaurs did survive. We just don't generally call them dinosaurs any more, but birds.