First of all, like centrifugal force, Coriolis force is an example of a fictitious force. Fictitious is an unfortunate name that makes it sound like a "pretend" force. Fictitious forces are real. They arise when you treat an accelerated frame of reference as inertial. That is, if you are stationary in an accelerated frame of reference, you pretend you are stationary in an inertial frame. This sounds unreasonable, but is done all the time. See my answer to Man inside an accelerating train carriage for an example and an explanation.
In this case, the Earth rotates once each day. Rotational motion is accelerated motion. But we who live on the surface of the Earth think of it as inertial, as stationary. The acceleration we experience from traveling in a circle once per day is small, so we usually don't notice it is present. But if you travel a large distance quickly, you would notice the effect.
It is simplest to start at the North pole. This place is stationary with respect to the center of the Earth. Look down on it from above. Imagine that it is covered with smooth frictionless ice. You push a rock south, say at 10 mph. From the rock's point of view, the ground is moving north at a constant speed of 10 mph.
Wait until it has slid to a point 100 miles from the axis of the Earth. Since there is no friction, the rock continues to slide straight south at constant speed. But the earth rotates under it. The ground travels eastward in a circle each day. The circumference is about 628 miles, so the speed is 628 miles/day or about 26 mph. The rock still sees the same northward component of the ground's velocity. But the ground has gained a sideways component too. The ground is accelerating in an eastward direction. As the rock continues southward, the sideways velocity component will grow.
Now switch points of view. You sit on the Earth and watch the rock go by. From your point of view, the Earth is stationary. The ground has no northward component of velocity. The rock has a southward component of 10 mph. The ground has no eastward component. The rock has a westward component of 26 mph.
From this point of view, the rock has accelerated. It started off going straight south at a constant speed. It has now acquired a westward component of velocity, and is traveling southwest. It has curved and sped up.
This is the Coriolis acceleration. It is real. A person who treats the earth as stationary is ignoring his own real acceleration. When he sees an object separating from his fixed position at a non-constant rate, he sees the object accelerating. He really will see the rock slide by at about 28 mph.
Switching viewpoints like this sounds incorrect, but it is a perfectly reasonable thing for him to do. He sees the earth is fixed. A nearby tree is fixed. Their velocity is 0. A nearby polar bear is walking at a constant speed. It is easy to keep track of the polar bear this way. It is harder to realize that he is traveling in a circle, the polar bear is traveling in a more complicated way than that, and to calculate the relative velocity between himself and the bear.
An object that is accelerating must be accelerating because of a force. The name of this force is the Coriolis force. You can see where the name "fictitious" comes from. But it is real.
If you want to keep the rock moving along a track that runs straight south from the fixed Earth point of view, you would have to make the rock accelerate eastward to counteract the westward Coriolis acceleration. You would have to exert an eastward force on the rock to counteract the westward Coriolis force. In that sense, the Coriolis force is real.
Note that the Coriolis acceleration is proportional to the southward velocity of the rock. With a large soutward velocity, the rock quickly reaches a region where the Earth is traveling eastward. The sideways acceleration of the Earth is large. From the fixed earth point of view, the westward acceleration of the rock is large.