1. All Mammals Didn't Survive
This needs to be said: all mammals didn't survive, just as all dinosaurs didn't die. Whole lines of mammals died out and the bird line of dinosaurs survived, doing extremely well.
2. Mammals Were Already Living in Pretty Crappy Environments
When you're a big plant eater, you tend to live in the best environments. The same goes for being a big meat eater. Those mammals? They weren't living in the luxury condos. Many were living up on mountains, near ice and snow, in deserts, and other more challenging environments. These mammals would have already been adapted to a harder, leaner life. They already had adaptation that let them survive hard times, such as droughts, freezes, and such. They may have even squirreled away food for those times as some modern mammals do.
3. Mammals Can Eat Utter Crap
Being a generalized small mammal has its advantages. When the going gets tough, they eat the tuff stuff. Look at rats and mice to realize just how many different foods that they can eat. If they can eat it, they will, which gives them a distinct advantage over the creatures who need more specific, more specialized foods to eat.
4. Mammals Turn Utter Crap into High Quality Food to Feed their Young
Dinosaurs are limited in the quality of food that they could feed their young. It couldn't be any better than what was available. In contrast, mammals convert low quality food into high quality food in the guise of milk. This gives their young far better calories, enabling them to feed their calories over time, which is especially important in a poor food environment.
Mammals also turn low quality food into high quality food via the placenta. Because mammals gestate, their young growing over time via calories provided by the mother, the mammals only need enough calories to start their young, supplemented over time by more calories eaten. In contrast, dinosaurs had to build up calories, enough to invest in eggs, and deposit these calories all at once. That's a big investment of energy at a time when the world just burned down.
5. Mammals Shield Themselves
As many mammals burrow, or find other small places to wedge into, they suffered less when the hell fires of Armageddon came. Not getting roasted alive helped their survival. In contrast, dinosaurs had nowhere to run, and the sauropods just couldn't dig fast enough. Poor guys.
Many types of mammals hibernate. When the great winter came, they already had the ability to hunker down and reduce calories over a long period of time.
7. Water From More Sources
Being small helps you to get water from more sources, such as dew and plants. Being big, you have to find big sources of water, and that gets tricky when the world has just ended. Dinos may have been good at conserving water (like birds), but they weren't godlike.
8. Emerging Food Sources
With the world freshly ended, lots of things died, which means free dinner for the bugs. So if you're the kind of creature that eats bugs, like many small mammals or birds, you get a big boon of food off all those other dying creatures. With all that biomass dying off and decaying, the critters would have had years worth of ants, termites, and other little critters to chew on, enough to tide them over to when things began growing again.
If these little mammals were anything like today's little mammals, they bred like rats. They would have multiple litters per year as soon as they had enough food, producing multiple generations before the larger dinosaurs had even accumulated enough food for one egg laying. As soon as their numbers could swell, they would have swelled, while larger creatures failed to struggle on.
Why didn't some mammal species survive?
1. They all lived in the wrong neighborhood. There no way of getting around a terminal course of urban renewal. So, think twice before mom badgers you (and your entire species) into living in the big city.
2. Too specialized. With their local pizza place destroyed, and the nearest place serving pepperoni being in Canada, starvation ruined a perfectly road trip before it even got started.
3. It's suddenly cold outside when it hasn't been cold before. Mom warned you to buy a coat if it got cold, but you thought that a jacket would do. Wrong. Go die of exposure.
4. In a mammal-eat-mammal world, the neighbor's new rude kids formed street gangs and pushed you out. In the competition for scarce resources, you lost.