One reason may be that natural fibres have space between them, and this serves to disrupt the propogation of sound waves. The natural variation within the structure might also contribute to this.
There may also be an element of absorbtion, whereby the fibres themselves are able to resonate fractionally, thus absorbing acoustic energy.
This is all frequency dependant, and the structure of any material's surface plays a role in reflection of sound waves.
Generally speaking the main factor that would contribute to the material's acoustic properties, would be the porosity within the material created by how the fibres are naturally stacked. Acoustic waves propagation through these pores would dissipate their energy into thermal losses due to viscous effects.
There are other factors of course, material elasticity, internal inertial losses due to material tortuosity etc. (Refer to Biot's theory for more details). Nevertheless, considering typical engineering implementations of such materials, for example acoustic panels for rooms, absorption due to viscous dissipation would be the efficient use of such materials. This however would make them more efficient to use for applications at higher frequencies (more than 800 Hz) . The following paper is a very good example of measured acoustic properties for a variety of natural fibres :
Article Acoustic characterization of natural fibers for sound absorp...
If you're interested in the way that different materials absorb sound, then the two properties of the material that contribute to this are its mass and elasticity. You can find tables of typical properties and absorption coefficients all over the web, such as:
https://cds.cern.ch/record/1251519/files/978-3-540-48830-9_BookBackMatter.pdf
and
http://www.acoustic.ua/st/web_absorption_data_eng.pdf
Are you interested in a particular type/family of natural fiber?
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