Why Does Wood Absorb Vibration So Well?
Wood is a Viscoelastic Material
Viscoelastic materials, such as wood, have properties from both viscous and elastic materials (hence the "visco" and "elastic"). When a viscous material is stretched, it resists flow because of internal friction. An example of a highly viscous material is honey; think slowly pouring honey out of a bottle. Contrast this with an elastic material. When an elastic material has an applied stress, it stretches, but immediately returns (or "springs" back) to it's original state after the stress is removed. An example of a highly elastic material would be a very stiff metal, such as steel.
A viscoelastic material (like wood) has both of these properties; there is an elastic component and a viscous component. Some viscoelastic materials, like rubber, have a larger "viscous component", so there will be a time delay (it doesn't quickly "spring" back) in returning to it's original state after a stress is applied. In this case, the energy dissipation will be greater. Wood, however, has a much larger "elastic component" than rubber which makes it much more stiff. Because of this, wood will react quicker to an applied stress, but thanks to it's relatively large "viscous component", it still dissipates energy .
This energy dissipating ability is called damping capacity. As we will see next, it is ideal for absorbing road vibrations.
The surprising truth about wood.
In a viscoelastic material like wood, there is energy being dissipated when it is loaded and then unloaded (i.e. when riding over a bump). Without getting too deep into engineering theory , let's look at the loop that has been graphed in green which illustrates what is known as hysteresis (shown below). The area of this hysteresis loop is proportional to the amount of energy dissipated . The more area the loop has, the more energy is dissipated. In our case, this energy comes from road vibrations and is actually being dissipated as heat. Yup, you read that correctly. Since the total energy doesn't just disappear (remember the 1st law of thermodynamics?), the energy in the frame is converted from vibration energy to heat energy. So does wood heat up from dissipating energy when loaded and unloaded, such as riding over rough roads? Technically, yes. But don't worry, the change in temperature is negligible over the entire bike frame.
Contrast this with elastic materials, such as steel, titanium, and aluminum metals. Metals do not dissipate energy nearly as effectively when loaded and unloaded. They have a very small viscous component to them and therefore hardly have any area within the loading and unloading curve. The curve shown below that has been graphed in red has no hysteresis, which is key to energy dissipation. Metals frames actually transmit or reverberate the energy (think of an aluminum tuning fork or bell: you strike it and it continues to ring). This means road vibration energy is not effectively dissipated as heat energy within the bicycle frame. Rather, it is transmitted through the frame into you, the rider.
Putting it all together
Wood's high damping capacity is what makes a smooth ride.
Damping is the conversion of mechanical (vibration) energy into thermal (heat) energy. The amount of energy dissipated this way is a measure of the structure’s damping capacity . Since wood frames have a high damping capacity (thanks to it's viscoelastic material properties), they are able to reduce "road noise" or vibrations from the road as you bike. Specifically, wood frames help to reduce vibration amplitude (the sharp and intense "noise" you feel from the road) and help to reduce the time for vibration to die out after an impulse (the "noise" you feel from the road dissipates quickly rather than carrying on). In contrast, metal frames (steel, titanium, aluminum) are made of highly elastic materials that have very low damping capacity; they do not naturally dampen vibrations nearly as effectively as wood does.
Fun Fact about the cellular structure of wood: wood’s viscoelastic behavior is due to it's lignin matrix . Lignin is an organic polymer that is important in forming the cell wall structure. The elastic regions of the lignin matrix respond instantly to the stress while the viscous regions respond more slowly.
Notes and References
 Viscoelastic materials have properties that are influenced by parameters such as frequency, temperature, and other influences. I have simplified this by not mentioning frequency and temperature impacts so that we can focus on describing and comparing wood as a viscoelastic material with metal as an elastic material.
 The point of this article is to give non-engineering minded people a better idea of why wood actually absorbs vibrations so well.
 OsinÌski, Zbigniew. Damping of Vibrations. Balkema, 1998, pg. 18-22.
 There is more than one method to quantify the damping effect, or damping capacity of a structure (a bike frame in our case). https://www.roush.com/wp-content/uploads/2015/09/Insight.pdf, https://www.ame.com/sawing-academy-articles/2017/08/14/minimizing-the-damaging-effect-of-vibration-and-resonance-with-stabilizers-and-dampers/