How was the Oh-My-God particle observed? How exactly was the Oh-My-God particle (ultra-high energy cosmic ray) observed and its energy measured?
 A: The OMG particle was observed by the Fly's Eye experiment located on Dugway Proving Ground in Utah. The Fly's Eye experiment was the first experiment to successfully employ the air fluorescence technique developed by Dr. Alan Bunner (Cosmic Ray Detection by Atmospheric Fluorescence, Ph. D. Thesis, Cornell University, 1967). 
The air fluorescence technique observes light emitted isotropically by nitrogen molecules in the atmosphere. The nitrogen molecules are excited by the passage of charged particles in the extensive air shower initiated when the cosmic ray particle collided with a nucleus in the upper atmosphere. This light is not Cherenkov radiation. 
Using simple arguments it can be shown that the total amount of light emitted at the peak of the extensive air shower is proportional to the energy of the primary cosmic ray particle. Measurement of this light is used to provide calorimetric determination of the energy of the primary particle. This is one of the strengths of the air fluorescence technique. Unlike ground array measurements, the air fluorescence measurement does not depend on detailed physics models or simulations. 
This is one reason that the Pierre Auger experiment and it's counterpart in the northern hemisphere, the Telescope Array Project, both depend upon air fluorescence telescopes to calibrate and cross check the measurements of their ground array detectors. 
In addition, air fluorescence observes the extensive extensive air shower development in the atmosphere. The ground based detectors can only sample the particle shower at a few discrete points on the surface of the earth.
The primary drawback of the air fluorescence technique is that it can only be used on moonless nights with good atmospheric conditions while ground arrays operate 24 hours and 7 days a week. 
I should mention that I am an author on the OMG particle paper.
A: Ultra-high energy cosmic rays create enormous cascades of charged particles as they interact and re-interact and re-re-interact in the atmosphere. This generates a lot of Cerenkov radiation and nitrogen fluorescence in the atmosphere and many of the charged particles reach the ground in a cone that may be miles wide.
So you build an array of ground station (typically water tanks a meter or two across instrumented with PMTs) for detecting ionizing radiation and a number of fluorescence telescope pointed at the sky over the array. The premiere example of such a facility is the Pierre Auger Observatory
Then you wait for a whopping big coincidence.
It also takes a lot of careful work and Monte Carlo simulation to tune the energy estimation from the results.
