How do woodwind instruments work (unlike brass instruments)? I can understand how brass instruments work because they are literally a long pipe, and all the notes are overtone based on the pipe length. But I can't understand woodwinds because there are so many holes on the pipe. It would make sense if opening a hole just make the length of vibrating air shorter so that the the pipe below the opened hole are "useless" and does not vibrate. But very frequently, woodwinds will open hole(s) and close some holes that are below (further away from the mouthpiece) the opened hole. This make no sense to me because that means holes lower than the opened hole do participate in vibration. So what is happening? 
 A: A simplified, yet very practical formulation of the phenomenon make use of the input impedance concept (i.e. the impedance which the player's lips, reed etc. sees). Its peaks (or throughs for flutes) correspond to the playable frequencies of the instrument. The thing is that the "end of the instrument at the tone-hole" usually doesn't have a desired value of terminating impedance and it must be corrected a bit. Hence the various fingerings, their combinations etc.
There are the time-domain concepts as well but they are usually quite complicated to be computed. Basically it boils down to a solution of Helmholtz equation (the spatial pressure distribution) with the Robin boundary conditions.
Note that the woodwinds use the overblow as well, but just once (the brasses overblow many times).
Source recommendation: 


*

*Fletcher and Rossing: Physics of Musical Instruments

*Chaigne, Kergomard et al.: Acoustics of Music Instruments
Further note: the distinction between woodwinds and brasses based on the tone-hole appearance isn't appropriate. There were brass instruments (labrosones) with the tone-holes as well. See e.g. Hornbostel-Sachs taxonomy for further reference.
EDIT -- Futher note II: You are a bit mistaken whn stating "and all the notes are overtone based on the pipe length". They depends on lenegth and the pipe varying cross-section which is very important for instrument building!!! Apart from the music acoustics references above you may see some pure physical acoustics to that point (e.g. Blackstock). The keywords are: waveguides, dispersion, Webster wave equation.
