The module will get colder at the far left (zero). Once the potentiometer is turned, the Crickit will cycle the power off then on in a ten second cycle. This is why the NeoPixels do not react immediately when you turn the knob, the previous off/on cycle must complete.
The Peltier will not get as cold as it did when it was directly connected to the power. The current is limited to one amp by Crickit so the unit will not draw all the current it wants. So if you want very cold, directly connect it to power.
If you want an environment that you want to measure the temperature (with the TMP36 sensor) and adjust the temperature, this setup will work for you.
A heat sink is usually a piece of metal, often with metal fins, used to dissipate heat into the air. They are used on a large number of heat-generating electronics. Most likely your computer has several, the main one for the CPU and others for the power supply system. Adafruit actually sells a 12 volt Peltier module with a heat sink attached although for this "Make It" guide we are limited by the Crickit for control which is 5 volts.
Placing the Peltier module's hot side on metal will help dissipate heat. If you plan to use a metal project enclosure, using the enclosure as a heat sink may be possible.
If you have access to scrounged electronics, you can often remove heat sinks from circuit boards. Heat sinks are available at many electronics stores also, both online and brick & mortar stores.
I found an old large chip heat sink in the junk drawer, but feel free to take any metal (usually heavier is better) to place on the hot side of the module to draw the heat away as we're looking to cool things.
A small fan also works, even better both a heat sink and a fan.
To get a good electronic contact between a heat sink and the cooler, consider some thermal tape, paste or thermal grease to place between the cooler and the heat sink. The material will allow for a good thermal contact between the two surfaces.