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
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
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.