The actual speed at which light travels is immaterial. The fact that it is not instantaneous is very important. Even though light travels very fast, the light from the subject or scene doesn't strike the sensor or film all at the same instant. Light reaches the camera from the subject in a stream of energy that is spread over a duration of time. For the time the shutter is open this stream of light is recorded in a photograph. If the scene changes over the course of the exposure, the shape of the stream of light reaching the camera during the exposure changes as well.
In physics there is often a phrase used to describe the way that light simultaneously demonstrates properties of both wave energy and particle energy: the duality of light. For the purpose of photography we normally treat light as a stream of photons flowing from the scene onto the sensor (or film). As they strike the sensor they are transformed into electrons in each pixel well that a photon strikes. As they strike film their energy results in chemical reactions to the grains of chemicals in the film's emulsion.
Why does shutter speed modify picture sharpness/detail?
The shutter time determines how long the stream of photons from the scene are allowed to strike the sensor. If things change position in the scene during the exposure duration then the light from the part of the scene that has moved will move across the surface of the sensor and fall on different pixels. If the camera itself is the source of the motion then the entire scene will shift and each point in the scene will fall on different pixels on the sensor. Whatever the source of the motion, the result is blur as the light from a single point in the scene is spread across multiple pixels. The longer the shutter is held open the greater the blur for the same rate of motion.
On the flip side of the same coin, the longer the shutter is held open the more light is captured in the photo. The more light that is captured by the sensor, the higher the proportion of electrons collected by the sensor from the light from the scene (we call this signal) will be to the electrons produced by the electronics of the camera that are also recorded along with the current from the sensor pixels. These stray electrons are what we call noise. Read noise is produced by the camera's electronics. Photo (shot) noise is produced by the random nature of light due to the duality of light. Those photon particles are travelling along a wave shaped path defined by the wavelength of each bit of light. The more signal (light) we have in proportion to the noise, the more detail we will be able to produce in our photograph. This is called the signal-to-noise ratio.
So a shorter shutter time minimizes the effect of motion but can lead to a loss of detail due to a poor signal-to-noise ratio. A longer shutter time increases the signal-to-noise ratio but can lead to a loss of detail due to motion blur.
Why do pictures get darker with faster shutter speeds, and brighter with slower shutter speeds?
Because the longer the shutter is held open, the more light is captured in the photograph. It is the same thing as turning a faucet on and off while holding a cup under the spigot. The longer the faucet is held open, the more water will be collected in the cup. The longer the shutter is held open, the more light particles (photons) will be collected by the sensor (or film).
Our eyes are always opened (when we are awake), but images are not "overexposed".
Again, the light is striking our eyes in a continuous stream, not in a singular instant. All of the light collected by our retinas over the course of a day, or a year, or our entire lifetime does not get transmitted to our brain in a single instant! The electrochemical signal from our eyes to our brains are continuously
changing as the scene in front of our eyes change.
(Note: the below was written before the question above was significantly re-edited into its current form)
Light is electromagnetic energy. As such there are two components that must be measured with regard to a photograph: field strength and time duration. Field strength measure how strong the light is over a specific area. Time duration measures how long that field strength is maintained.
It is just the same as any other form of energy. If one were to apply a constant force against a body the body would accelerate. The longer that force is applied the longer the body will accelerate and the faster the body would be moving relative to its beginning state.
A piece of photographic film collects information about the energy falling upon it in the form of light. The longer the shutter is left open the more information is collected. If a shutter is left open twice as long it will collect twice as much information from that light assuming the strength of the light is constant.
The problem in photography is that the light is often not constant. As things in the world in front of the camera move the field strength of light over any particular point of the film or sensor changes. As long as the shutter is open it continues to collect information about the light falling upon each point of the film or sensor. If something in the view of the camera is moving the information about all of the positions it passed through during the time the shutter is open will be recorded. Instead of being recorded on the same spot on the film or sensor the image of the moving subject will be spread across the area over which it moves. This will result in blur. Even if nothing in front of the camera moves, if the camera itself moves the same thing will happen. Light from a particular spot in the scene will be spread over the area of the film or sensor upon which it falls as the camera is moved.