And our ancestors at the time were the early mammals - smaller, rat-like creatures, and a few other species. They were mostly scavengers who were better able to survive the difficult conditions, while many other animal and plant species died.

As things got better, without giant lizards stomping around everywhere, those mammals were able to thrive, diversify, and spread all over the world into the wide array of mammals we have today. Including the first primates, somewhere around 55 million years ago (well after the meteor extinction event), who would eventually (millions upon millions of years later) give rise to us.

Also worth noting a slight variation on this theory that is the currently held belief.

Unlike most modern reptiles, dinosaurs were actually warm blooded. Opposite of what they used to believe this actually played against them.

It used to be taught that because they were cold blooded they couldn't handle the cold and that's why mammals dominated afterwards. But this as it turns out is not true, they were mostly warm blooded like modern birds.

Being warm blooded means you have a MUCH higher and more demanding metabolism. Snakes for example only need to eat once a month or so to be completely healthy and they can go much longer if needed. Most warm blooded creatures need to eat pretty much daily in order to remain healthy.

When the meteor hit and the ice age started food became much more scarce. Gigantic warm blooded animals simply couldn't eat enough food to survive. Meanwhile some larger reptiles like crocodiles survived precisely because they don't need to eat much, and can go into what is almost a hibernation like state in extreme cold where they can go a really long time without eating. They just stick their noses out of the water and breathe really slowly and can survive even when the water is frozen for several months.

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The Fungal Infection Mammalian Selection hypothesis says that the killing mechanism was unchecked fungal infections as the temperature dropped. Modern cold blooded creatures suffer from bone infections if their body temperature stays too low.