The statement should be understood in an asymptotic sense, just as its analog for integers (the probability that any two randomly chosen integers are relatively prime is 6/Pi^2).
For an illustration, consider a simpler question, what is the probability that a randomly chosen integer is divisible by 5? Among the numbers {1, 2, 3, ..., n}, there are Floor[n/5] multiples of 5, so the probability to choose one at random is Floor[n/5]/n. The limit of that probability as n goes to Infinity is 1/5.
I don't have a link to the full Collins paper, but here is a
preview. Also, see this
paper for another proof.
The following experiment generates 10^5 pairs of Gaussian integers and compares the empirical probability to the theoretical value:
In[6]:= r:=RandomInteger[{-2^128, 2^128}, {10^5, 2}]; N[Total[Boole[CoprimeQ @@@ (r + I r)]/10^5]]
Out[6]= 0.66395
In[7]:= 6./(Pi^2 Catalan)
Out[7]= 0.663701
