Get together all your parts (see BOM). You also need a PCB: Download the Gerbers and get them produced yourself (at least for the moment). Note that the PCBs have as of now (Christmas 2014) a couple of (fixable) bugs. Alternatively, you can build the project on a breadboard or perfboard - use through-hole-components then. If you use a PCB, make sure to read an SMD soldering tutorial and use the right tools.
Start by soldering the three 100nF caps on the topside of the board (C1,C2,C3).
Solder R5, this is a 270 Ohm resistor (you may also use another value between 150 Ohm and 300 Ohm).
Solder R1 and R5, those are 100 kOhm resistors.
Solder R2 and R4, those are 10 kOhm resistors.
Solder the transistor Q1. Note: If you have a logic level FET with a lower on resistance than the BSS138, use that!
Solder the WS2812b LEDs. Note: They have a little cutout that must be oriented to the side of the capacitors. If in doubt, check the datasheet!
Turn the board over. Solder C4 (100nF) and R6 (10 kOhm).
On the top side of the board, place the switch and solder.
Solder the DHT11 sensor on the bottom side as shown. Mind the direction of the pins! Note: I found that this placement is not ideal, as the heating from the gas sensor influences the temperature/humidity sensor. Instead is soldered the DHT11 on some wires and the wires to the board until I come around to redesign the PCB!