It is interesting to note that, in the Wikipedia article, it states that
The amino acids were racemic (that is, the chirality of their enantiomers are equally left- and right-handed), indicating that they are not present due to terrestrial contamination".
This implies that, if a 'life form' did create these proteins, they are fundamentally quite different to life on earth. (Particularly in light of the 70 unique amino acids they described!).
We cannot ever 'know' how they were made, unless a pretty revolutionary discovery is made that undisputably confirms evidence for extra-terrestrial life (although this is the best evidence yet!), so we are left to speculation.
The nucleobases either formed 'randomly' in one of the countless celestial events (although the chances of this seem pretty remote given the complexity of the molecules and the homogeneity in the sample), or they were made by extra-terrestrial 'life' (much more exciting, and indeed probably more likely, as we will see). (Or the 3rd option: intelligent design, however this is dramatically less probable than either of the other two).
The Miller-Urey experiment to which you refer has been subsequently enhanced, and evidence for the "catalytic potential of cosmic dust" is now reasonably robust (Hill, 2004). They demonstrate that simple nebular iron silicate catalysts can yield methane and water from CO₂ and H₂, and ammonia from N₂ and H₂. Allowing these to react yields nitrogen-containing organics such as methyl amine.
It is not a huge jump to suppose that with enough time, mass and energy, these reactants could combine and form a molecule (chain of amino-acid-like molecules - a protein) capable of faithfully replicating itself, and thus initiate life. There is no reason to suppose that Earth is unique in this respect; there are billions of suns in our galaxy alone, and billions of years of time for these reactions to take place.