gluon,
No, 'they' did not state the 'source' was 7 billion years old, that was your speculation. The source of the peak in the radio wavelength is unknown, all they did was subtract estimated foreground contributions and were left with too strong a signal at that particular wavelength. Since the signal seems to be diffuse and not associated with known nearby sources, it is assumed to arise from the early universe, when the universe was less than half its present age. That doesn't mean the signal is 7 billion years old, it means that the signal, if real and not an error, is at least 7 billion years old, meaning it likely comes from the early universe when galaxies were first forming. Early in their formation, galaxies seem to be much more active than later in their evolution when they are more mature. Quasars and other AGNs were common in the early universe, and if the Big Bang is true, our universe was a much smaller place early in its existence. The signal detected was an excessive hiss of white noise at around the 3 GHz wavelength, not an explosion or something that happened 7 billion years ago.
From what I understood the noise comes from the "outside" fringes of space, the other direction of where they were trying to get information from the direction of the "big bang" (perhaps just the super-primordial-"quasar" which formed most the the matter in our "known" space).
Outside the known universe. "Big Bang" direction is IN (indeed the center) of the known universe.