Good question. Your last sentence hits the nail on the head. To paraphrase: "When we see star light that we know is a billion light years away do or can we determine what was happening a billion years ago?" The answer is yes.
To set the scale, the age of the universe, as we know it and according to current theories, is 13.7 billion years or so old. Modern telescopes can see light that was only a few 100 million years after the big-bang.
Two points in your question:
1. How do we know how far away a star is? The answer lies in the Hubble relations among distance, the receding velocity of a light source, and the Doppler shift of wavelengths of certain discreet transition lines in atomic spectra. http://en.wikipedia.org/wiki/Hubble%27s_law
By measuring the observed wavelength of certain well known lines in a stellar spectrum we can then determine how fast the star was moving away from us. On a big scale, that tells us roughly how far away the star was.
2. How can we see such faint stars? We build bigger telescopes. The light gathering ability of an optical system grows as the diameter squared. The pupil of the human eye is about 5 mm in diameter. Modern telescopes have diameters of 10's of meters, those of radio telescopes are hundreds of meters. Since the mount of light falls off as the square of the distance, how far you can see goes as the diameter. You can do the arithmetic. Furthermore a telescope can integrate the light over many minutes of exposure and enhance the signal. Further furthermore, modern electronic sensors are much more sensitive than the human eye, and also make time integration more easy. We've come a long way in observational astronomy since Galileo.
(published on 10/09/11)