can this be used to do what I want?
You didn't mention whether you want control over each LED independently, or not, so it's not possible to answer this question.
Can one Arduino handle that much power or do I have to use relays?
No and no. Notice down the right side of the page are the components used in the project, including the LED strips and MOSFETS. The MOSFET is turned on and off by the Arduino. It switches the large amount of current that the LEDs need. Like a relay, but much faster, quieter, and cheaper.
If relays are required can the Arduino outputs switch relays?
They aren't, but if they were the answer is "it depends". Typically, no, the Arduino toggles a transistor which toggles the relay.