For most applications this circuit can be simplified a little bit. The capacitors are only needed if your load circuit is sensitive to small power fluctuations. So in many cases, these can be eliminated. The variable resistor R2 is useful if you want to be able to power multiple different devices. But if you are going to use the power supply exclusively on one device you can replace it with a fixed value resistor. Wire the circuit as shown with Vin connected to the power supply and Vout connected to the circuit that you want to power. The regulator will bring down the output of the power supply down to the value that you set.
Depending on the power rating of your circuit, you may need to add a heat sink.
My son's swing normally runs on four C size batteries. So I found an old power supply with a 9V 1000mA output. I figured that would be enough to replace the battery pack. Then I soldered together the LM317 regulator circuit with a 220 ohm resistor for R1 and a 820 ohm resistor for R2. These resistor values give an output voltage of 5.9V. (It would have been ideal to use a 240 ohm for R1 and a 910 ohm for R2 but I didn't have those values on hand) This output is still well within the operating range for a four cell battery pack. Anything between 1.25V and 1.5V per battery will usually work. Since the electronics on the swing just consists of a motor and a speed controller, I decided that the filtering capacitors weren't unnecessary and I left them off. See the following steps for the best methods for connecting everything together.