If you are new to Arduino and are wondering why more than a few LEDs or other components like motors won't activate when functions are called in the code, its because each output has a current limit of 40mA. In other words, a component cannot draw more than 40mA of current from each channel. To refer to the water analogy of electronics, the "pump pressure" is 5V, and the amount of water is the number of electrons (measured in Amps, or in our case, a much smaller amount - milliamps, mA). To accommodate a load that requires more current than 40mA at 5V, we will use our microcontroller to control a transistor, which will provide a component with power from an external source (the battery).
Without getting too technical, its worth knowing that individual strips are made up of 3 LEDs in series which can be cut with clippers at any junction. If you want to cut the strip at any point, just be sure to leave connection points on each halve. To understand how the RGB LED strip can be powered with 9-12V, you need to know the difference between circuits in series vs. parallel (this page has a simple explanation with great illustrations, and there is a popular Instructable that covers wiring LEDs in series & parallel). Basically, when active components are connected in series, their supply voltages are added together. For example, since an average RGB LED requires 3.3 V and 60mA (at full brightness; each color channel draws 20mA, so R-G-B all on at same time is 20 x 3 = 60mA), each strip of 3 RGB LEDs will require approximately 9.9V (the strip I'm using from Jameco can be powered between 9-12V. Be sure to look at your product's datasheet to prevent frying your components. Not all RGB LED strip is powered in the 9-12V range, such as Adafruit's digitally addressable RGB LED strip). One more thing, these strips are "common anode," meaning the LEDs share a positive terminal (read about anode vs. cathode).
Perhaps the greatest take-away is the power limitation of the Arduino. The next section which shows how to use a transistor can be applied to all sorts of other components (ex. motors, solenoids, servos) that require more than 40mA at 5V.
Note: The same concept applies to NeoPixels, but they require a lower voltage (5V). If the power source is lets say a 9 or 12V battery or charger, the power must be regulated.
Note2: The water analogy turns out to be a pretty poor way to visualize what actually occurs at the electron level. William Beaty explains how transistors actually work.