The capsules designed to reenter the atmosphere have to slow down from about 8 km/s to zero by the time they get to the ground. They actually don't use the part that looks like a cone to do that. They all have flat bottoms that they face into the wind to do that. If you compare the Dragon capsule from your link to a Soyuz capsule, the Orion capsule, or the Space Shuttle, their shapes are very different, but they all have a part that is more pointy (though not very much, in the case of the Soyuz), and a part that is flat. The pointier bit is used when they are launched, to lower the friction of passing through the atmosphere (the Soyuz is launched in the middle of the rocket, and so it doesn't need a point). The flat part is used to come back.
The black lines in the picture show where the front of the blast wave occurs when these shapes pass through the air at very high speed. The blast wave is what is slowing the capsule down and you want it to be as big as possible. A sphere is actually the shape that makes the biggest blast wave - in fact, it works too well to be used on a capsule with people inside. It slows down so fast they feel crushed. All the shapes used are partly spherical, on the part that is facing into the wind, behind that they taper in, but how much is optional. The Space Shuttle was an exception - it had only a very slight curve on the bottom because that was best when the overall design was considered. It worked because the surface area it had for slowing down was so large compared to the size of the whole spacecraft.
Notice how the blast wave doesn't touch the craft, it moves along in front of it. The craft moves so fast there is a bubble of air trapped in front of it, compressed between the outside face of the blast wave and the spacecraft. That bubble is also very important. When air is compressed, it heats up. The faster it is compressed, the more it heats. The fastest compression is on the face of the blast wave, and so that part is hottest. The air bubble also gets very hot, but not as hot as that, so it protects the spacecraft from the worst heat.
And see the fuzzy section behind the craft? That is an area where the air is thinner, because the spacecraft is punching a hole through the air, and it takes some time for that hole to fill back in after it has passed. That creates suction pulling on the back of the craft, and that slows it down too.