1 - The LED strip says that it needs 12 volts at 3 amps. You are providing 12 volts at 1 amp. I'm tempted to ask, "Do you see a problem?" but you obviously don't. So get at least a 3 amp supply, already.
2 - You are driving the transistor bases directly, rather than through a base resistor. This is causing the Arduino to try to provide way more current than it can, so the Arduino is getting hot.
3 - If you do provide a base resistor which makes the Arduino happy, the transistors will not be driven hard enough to work without a heat sink, and they will get very hot and probably fail if you do it long enough.
4 - Regardless of whether the transistors fail, they will not be able to drive the LEDs to much brightness. A single transistor will not have enough gain. Sergei Grishin has the right idea, but make sure that the MOSFET can drive one amp reliably (some cannot, although most can), and that the maximum "gate threshold voltage" - Vgss - is greater than the output voltage of the Arduino pins. This will not be a problem if your Arduino is putting out 5 volts, but some MOSFETs will not work reliably with 3.3 volts on their gates.