Why do we think of light as a wave?

Light is a wave - an electromagnetic wave. Radio waves and microwaves are also electromagnetic waves, they just have different wave lengths. Wikipedia has a nice picture showing the electromagnetic spectrum

why does it have a wavelength...

It has a wavelength because there is physical space between the peaks of the waves - it is a real, physical, wave. Just like water waves and sound waves, you can do "wave things" to light waves, such as send them through diffraction gratings and see the interference.

and what creates its wavelength

Whatever creates the light gives it energy, and the wavelength is proportional to that amount of energy:

$$\lambda = \frac{hc}{E}$$

Does the light vibrate too? If so, then how?

Sound waves are energy waves that compress matter - you can't have a sound wave in a vacuum. Light waves are energy waves too, but they don't need matter to go forward. That's why we can see sunlight, but we can't hear the sun. (And that's why in space, no one can hear you scream.)

This subject can get really complicated really fast. Because although light is a wave, it is also a particle. A lot of really smart people have been scratching their really smart heads over that, and will be for a long time.

Light is a electromagnetic wave. We talk about the dual-nature of light since in some instances it displays properties of a wave and other times as particles, each in a different state of quanta.

When a metal is heated it will emit photons as the electrons shift their energy levels. Since electrons are emitted from a metal at a certain frequency of light it indicates that light consists of a particle called a photon. Increasing the intensity of light increases the rate of electron emission. This is not a property of a wave.

However when light hits a transparent object at a angle it will be refracted (except at 90 degrees). This is a property of a wave.

Therefore light is seen as electric waves and magnetic waves oscillating at right angles from each other. Light is always either a wave or a particle but never both at the same time. Welcome to the dual-nature of light.

I want to give you an example of how light can created. This has been helpful for my students when it comes to visualizing light as a wave. This is not the only way that light can be created, but it's a way that is a little easier to visualize and may help you understand why it can be seen as a wave.

Imagine an electric charge. This charge creates an electric field around itself, which is what allows it to put a force on other charges.

If we move that charge, the electric field associated with it will also move. However, and this is really key: the electric field does not adjust instantaneously. It takes time for the field to "catch up" with the new position of the charge. There's a great simulation of that at this link. It's a little exaggerated, but it'll show you the basic idea. You can see that wiggling the charge creates a disturbance in the electric field that looks very wave-like.

There is also a magnetic part to this, which is more complex and is not shown on that animation.

Moving charges are how we create radio waves, which are a form of light. (They're not the visible light that you were referring to, but there are a lot of different kinds of light.) Radio waves are made by moving electrical currents up and down the broadcast tower via a circuit that's designed to do so.

It took a while for people to realize that radio waves, visible light, x-rays, and many other things are all really different forms of light. Once we realized that, it was clear that we could represent visible light with a wave - an electromagnetic wave, with a wavelength, frequency, amplitude, speed, and all the things that waves normally have. The wave is not "light vibrating," the light itself is vibrations in the electric (and magnetic) field.

I should also mention that @Robert is correct. Light can be modeled as a wave because it behaves like one under certain circumstances. It can also be modeled as a particle (the photon) or as a collection of waves called a "wave packet."

I hope that helps!