A planet may look very solid and hard from our human perspective, but in the large scale it can be compared to a ball of slushy water suspended in midair. An asteroid, sufficiently large in comparison, can not just crack the planet, but actually smash it into it with such force that the released kinetic energy turns both bodies into smaller droplets, just like hitting the floating ball of water with a baseball bat.
What afterwards happens to the droplets depends on a lot of things, but most likely the majority of the fragments will reattach, melt together and form a new planet. The remaining fragments might create a ring around the new planet or even form one or several moons, such as this computer simulation below shows.
This is in fact what is now believed to be the origin of our own Moon: a smaller planet called Theia smashed into the original Proto-Earth some 4.5 billion years ago, breaking it into glowing droplets that later formed our current Earth and also the Moon.
If that would happen today, the Earth would pretty much turn inside out and after the carnage, not a single shred of our civilization or anything resembling our home planet would remain. The oceans would evaporate and vent out into space, forming a spectacular, sparkling ice ring, and most likely eventually falling back as thundering, dirty icebergs crashing down onto the planet with devastating force.
The new planet would probably end up having some kind of atmosphere, initially perhaps resembling the atmosphere of Saturn’s largest moon Titan, consisting mainly of nitrogen and methane. The new Earth would not look like a blue marble from space, but sickly brownish yellow, the atmosphere more or less opaque and filled with smoke and murky clouds, only occasionally revealing a bit of a scorched and black surface, with rivers of lava faintly glowing in the night.
The new planet would initially be fully exposed to the Sun’s deadly radiation, but eventually the heavier material, mainly iron, would settle back into the planet’s core and start rotating, creating a new magnetic field that would shield the surface from the solar wind.
As the comets consisting of water and soil from the old Earth’s oceans would come crashing down with regular intervals, they would bring back water to the planet. Carbon dioxide would be released, again reinitiating the atmospheric greenhouse effect. New continents would begin to form and the new water would eventually collect in basins to form new oceans and lakes.
Evaporation would create new clouds, rain would start to clear out the atmosphere from airborn dust so it becomes transparent once more and the sky would again start to turn blue, at least where not shrouded by the deadly smoke and gas from the remaining lava rivers and erupting volcanoes.
After a billion or so years, when the new heavy bombardment has ceased and the planet has been resolidified, life might eventually emerge again in oceans, as microbes, algae and possibly more advanced shapes conquering land once again. But whatever it would be like, it would not have any resemblance to what exists today.
There would be absolutely no remains of our cities and infrastructure, not even fossils from the original biosphere. The only physical evidence that we ever existed would be the handful of slowly receding probes leaving the Solar System.
Should we be worried? I’d say no. The risk for such a cataclysmic event is very, very, VERY low. There are no known asteroids or planets in the Solar System big enough and in an excentric enough orbit for anything like that to happen. Rogue interstellar planets might of course exist, but the probability of one finding its way to our home and hit the Earth is so astronomically slim that it is almost non-existant. Furthermore there would not be a single thing we could do to prevent it, so, well, why worry.