Well I googled for http://www.fabiobiondi.com/blog/2009/11/control-your-arduino-using-an-infrared-remote-and-the-ik38khz-sensor/ . If that is the project you are talking about, I can't find a wiring drawing in the page and the picture quality is not good enough to tell what is going on. If your question is how do you properly size a series current limiting resistor for an LED, then maybe I can help a little.
First a current limiting resistor is required for leds because once an applied forward voltage equal or greater then the Vf rating (diode forward voltage drop) for the led, it starts to conduct and will effectively act like a short circuit unless something external to the led limits the current into this quasi 'short circuit'.
Moving on to sizing the resistor, one needs to know the following information before the calculations can be made.
Vf rating of the led (green are typically 3.5vdc I think)
Desired continuous current for the led (20ma is typical max)
Value of source driving voltage (Most Arduino output pins are +5vdc.
Resistor in ohms = (driving voltage - Vf) / .020 amps = 75 ohms
A red Led usually has a Vf value of 1.5vdc so ohms = 175 ohms.
Now frequently you will see larger value resistors used instead of what I show here. That is because a typical led (with 20ma max rating) will display light form anywhere from 1ma on up to 20ma, and it's hard to tell a real difference in brightness from around 10ma on up, so why use more current then needed. I think most arduino boards use a 1,000 ohm resistor driving the on board led at 3ma.
Now as to you 10,000 ohm resistor, I wouldn't think that would allow enough current to light the led, note that the resistor can't 'stop' current, only reduce it such a value as to not generate light in the led. Resistors just participate in Ohm's law, they can't start or stop anything. But you say your led lights? Hum, is it really a 10,000 ohm resistor? Did you measure it with a ohm meter? Did you properly read and decipher the color coding bands on the resistor? Only you can answer those questions.
The important point is you should always insure that you aren't drawing too much current from an arduino output pin as that can damage the pin and the circuitry you are driving. AVR pins are rated at absolute max on 40ma, recommended 20ma, and a max rating of 200ma for all output pins combined.
Hope that helps.