ZLC Controller reported a language issue with a pilot while attempting to issue a reroute; which resulted in an airspace deviation. Noise and congestion in the control area was cited as contributory.

Narrative

My first shift back to work; I was asked by TMU to reroute Air Carrier X going to Mexico. I issued the clearance to Air Carrier X the first time and he did not understand. There was definitely a language barrier. I repeated the clearance again to the pilot. The routing was a change that would occur in Texas. After the third attempt I believed that the Pilot read the clearance back correctly. I was then later informed that the pilot didn't follow the clearance completely and had turned off course 1 DEGREE; causing a deviation. Air Carrier X proceeded direct VUH (130 degree) instead of CZI (129 degrees). Not only because of ERAM but because of such a small turn the aircraft left my sector in a flat track. I am definitely concerned that I was asked to give a clearance that I was not familiar with; and one that did not need to be done until Houston Center; and that was issued to a Pilot that I definitely had a language barrier with. I definitely feel that while ERAM was being tested; the noise level was incredibly high and the body count in the area had doubled. I definitely think that this working environment was a factor in not catching the Pilot's error in his readback the THIRD TIME.