How does the Eco-Cooler air conditioner really work? This article describes a device for developing countries which apparently cools the air by 5 °C without electricity. According to its YouTube video, it works by lowering the pressure to drag in the ambient air. Wouldn't the ambient air be at the same temperature?
Does the constriction compress the air and temporarily increase the pressure and so temperature, which then dissipates heat because it is hotter than surroundings; then after passing the constriction, the pressure returns to almost ambient but the temperature drops to below ambient?
How does the Eco-Cooler air conditioner actually work, if at all?
 A: Here is a simple experimentalist's answer:
a) Tin huts in the sun get extremely hot, as anybody who has left a car in the sun will know, much hotter than the outside temperature. 
b) People in Bangladesh will have about the same IQ as people in other countries, therefore these tin huts will have at least two windows for a cross current in an effor to lower  the inside temperature drawing the outside air in. ( I was thinking of chimneys, but these are for cooler countries). Leaving windows open is what one does to keep a car cooler in the sun.
The smart thing is to have the windows north/south, because even on windless days the south side outside the house heats more than the north,  and more than inside the house :  a natural convection happens between the two windows: hot air rising on the south pulling cooler air from inside the house, and the air replaced by the north window, with the cooler air of the shadow of the house.
If you watch the video there is no sun on the window used, so it must be the north side window.
This natural circulation brings the tin hut to a lower temperature but still not equal to the ambient outside air in the shadow. 
The cooler, replaces the north window. What happens is that the air pulled in because of the open south window goes through the smaller openings and acts as a fan . I.e. instead of an electric fan, and everybody knows that it does not lower the room temperature, this  is a passive fan, as explained in other answers. 
The room as a whole will reach the same temperature as before the installation, but the people next to the cooler have the same feeling as sitting in front of a fan or using a hand fan. The video shows them next to it. In addition, the temperature next to the cooler will be the outside temperature in shadow, which is certainly lower than the one in  a tin hut.
In the video they also say that how well it works depends on the wind. On a windy day convection wind currents are  much stronger.
A: I can't rule out the possibility that it gets more comfortable inside with the "eco cooler" installed, but the explanation that they give is utter nonsense. They show a thermometer "before" and "after", but they don't tell what the "before" condition was: open window? glass panel? wooden shutter?
CuriousOne already hypothesized that it works by blocking sunlight and still admitting air. Could be. We can't tell from the video clip or article.
They offer two "explanations", which I paraphrase:
(1) "When you blow a jet of body-temperature humid air on your skin, it feels cooler than the same air volume at low speed";
(2) "As hot air passes through, the bottle neck compresses and cools the air".
The explanation for the effect in #1 is that the jet will drag dry air with it, thereby amplifying the air volume, and the high speed will lead to better evaporative cooling from the skin.
The "explanation" in #2 has several problems. First, terminology. In physics, the term "compress" refers to reducing the volume of a fixed amount of matter. If you compress a given amount of air, it will generally increase in temperature. Here, they seem to mean that the cross section of the air stream decreases along the line of the flow. 
Bernoulli's principle will tell you that the accompanying velocity increase must lead to a drop in pressure, compared to the pressure in the wide part of the bottle. The pressure drop may even lead to a temperature decrease of a fraction of a centigrade. However, when the air slows down as it mixes with the indoor air, the pressure will increase again and the temperature will increase again (also by $\ll 1$ °C) as you follow the air-flow path.
How much is $\ll 1$ °C? Let's say that the highest velocity involved is $v=5$ m/s (that's a nice breeze). Bernoulli: pressure difference is $\Delta p=\rho v^2/2=16$ Pa. Adiabatic expansion of gas: $pV^\gamma=\mathrm{constant}$, with $\gamma=1.4$ a property of air. With the universal gas law ($pV=nRT$) and a first-order approximation, we can derive that the temperature drop in at the highest velocity point is
$$\Delta T \approx T\frac{\Delta p}{p} \left(1 - \frac{1}{\gamma}\right).$$
With $p=10^5$ Pa as the atmospheric pressure and $T=300$ K as the absolute temperature, we get $\Delta T\approx 0.014$ K. (And even then, heat exchange between a stationary object and a high-speed gas that has a lowered temperature due to its high velocity is a difficult topic that could actually to heat transfer in the opposite direction of what you'd expect, in some cases.)
Another misconception here is the implicit assumption that the volume of air (per time unit) passing through this device is the same as if there had just been an open window. If you blow with your mouth open, your lungs need to produce less pressure than if you blow the same volume flow rate with your lips pursed. But with the wind, the available pressure is a given; placing a flow restriction in the wind can only reduce the flow rate, not increase. In the bottle necks, the air flow velocity may be slightly higher than if you had had an entirely open window, but the total volume rate of air passing through will be much smaller. How much, that depends on how the air leaves the building and of course the direction of the wind outside.
A: Without any outlet, under sunlight heat, the temperature inside the house will be high and higher than ambient temperature. The pressure will be high as well. 
If we keep let air flowing into the house with "eco-cooler", air mass accumulates and the pressure increases until air cannot flow into it. 
So in order to keep it working, there should be a way to remove the air inside the house. I would guess on the house roof, there is a "chimney" that can pull out the hot air out due to pressure difference or buoyancy flow between the house and the surrounding. 
Putting a window as air inlet on the wall can establish this air circulation. Inlet air will be lower in temperature and air flow will lower the wet thermometer even further. If we can put a basin of water at the inlet, it will be more effective. The plastic bottle stuff and Bernoulli equation is misleading. After air leaves the bottle, it will be the same or worse than it is before. Otherwise it will violate second law. Further bottle reduces flow coefficient.  
A: The idea of eco cooler is that the air cools down while expanding.  The problem is that air warms up while compressing!  So in the eco cooler as shown, air is first warmed up and then cooled down, bringing us back to initial temperature.
Eco cooler would work if you put thermal exchanger between air compression and air expansion.  So first you compress and heat the air, then you cool it back to ambient temperature (out of the house) and then you expand and cool it again (in the house).  This is exactly how air conditioner works!
The problem is how to keep the air compressed while in thermal exchanger.  Thermal exchanger can be visualised as a very long pipe in contact with the outer air.  It is impossible to done that without some mechanical aid and energy consumption.
The bottom line: there is no free lunch.  You cannot make air conditioner that does not require some energy input.  Eco cooler is fake.
A: First off, the fact that the board actually blocks the sun light going into the house may have cooled down the house itself (same effect as a solar screen). Since this question is about how the pressure and temperature will be changed after installing the Eco-Cooler air conditioner (the bottle board solely), I will give the following analysis.
From the Gay-Lussac Law that 
\begin{align}
\frac{P_1}{T_1}=\frac{P_2}{T_2},
\end{align}
the ratio between pressure and temperature is a constant for a given approximately fixed volume. When one installs the bottles on the windows, the shape of the bottles increases the air pressure before entering the room. This can be understood in the following manner: we assume the wind is entering an almost closed house that every cross section over the course of the bottle pipe is approximately under the force balanced/equilibrium condition that $$F_a=S_aP_a\approx S_bP_b$$ for two arbitrary cross sections $S_a$ and $S_b$. Since the bottleneck area is the smallest, pressure there may be the biggest before entering the house. In this process, however, the temperature may not be changed since it is contacting with the outdoor environment constantly. 
When the air blows into the room, the pressure gets decreased immediately to the normal or even maybe below-normal (depending on the actual room pressure) atmosphere pressure as there is no bottleneck-shaped pipe limiting its volume. As a result of the equation given above, the temperature goes down immediately inside of the room. 
One important note regarding the other answers and the valid condition of equation (2) above -- I have noticed other answers have been focusing on the opening of chimney and other windows, but here I don't need to assume that condition, and indeed I think the other outlets of the house should keep closed to prevent heat exchange from those openings. Firstly, we shouldn't focus on whether there is a chimney or outlets on the other side of the house to explain the temperature change due to the bottle board installation. Because before or after the bottle board is installed, the other chimney or windows are always there if there were any, they shouldn't be the cause of the change of temperature -- it is the installation of the bottle inlets generate the temperature change. Secondly, since the house is relatively large compared to the openings from the video, air flow will experience a friction effect when it enters the house (all the other answers haven't considered this effect). In other words, you can imagine the house is approximately a closed volume that it will give a resistance on the entering air and reduce its entering speed. Therefore, the air flow going through the bottle pipes will be compressed at the bottle neck position and the speed of entering the house is lower than the case that it enters an completely open space. This validates the condition of equation (2) which is any cross section of the air flow over the bottle path is approximately in a force balanced/equilibrium condition. Thirdly, a chimney may help to let warm air go out and the houses shown in the video are made by wood and are not air-tight indeed, but it is actually important to keep other big windows of the house closed in practice to prevent the hot air entering the house to raise the temperature again! The video shows the room temperature can be $5^\circ$ lower than outside. If you keep other big windows open, it is very easy to rebalance the room temperature back to high again. This is the same common requirement as we turn on air conditioner in summer, and makes the point 2 above even valid. Obviously, other answers may have ignored this common knowledge -- instead of analyzing how the bottle board helps but to argue about there must be openings to let air flow freely go in and out of the house to make the cooling process possible if at all. 
Feasibility and conditions to make it work: We see from the video that there is a $5^\circ$ temperature difference. We can assume that the outside temperature is about $30^\circ C$ or $T_1=303K$; and the temperature inside is about $25^\circ C$ or $T_2=298K$. Therefore, the pressure raised at the bottleneck relative to the normal house pressure is 
\begin{align}
\eta=\frac{P_1}{P_2}=\frac{T_1}{T_2}\approx 1.017,
\end{align}
which is about $1.7\%$ of pressure increase. From equation (2), since the bottleneck cross section is a lot of smaller than the intake area, the ideal pressure increase can be a lot more than $1.7\%$ when Equ. (2) is completely an equation. Considering Equ. (2) becomes a complete equation only when the bottleneck is completely closed from the house side, which is not totally true, and the house is constantly exchanging heat from other non-ideal channels with the environment, we may find the $5^\circ$ temperature decrease is possible from a rough estimation. To make the Eco-Cooler work well, it is crucial to have a good insulation condition of the house and make sure all the other windows/openings of the house closed to make the constrain well satisfied. However, if there is no air flowing into the house from the bottles, this Eco-cooler may not work well from the pressure-temperature transitions, but it may still work to some extent by blocking the sunshine from shining into the house. 
A similar rule governs the case that when you evaporate water inside of an open room, in which the volume of the water vapor is increased from water and the chemical energy of water vapor from liquid water is also changed so that in the end water vapor will absorb heat from the air. Hopefully this helps your understanding on the power of physics laws. 
