The 'galaxy teeming with life' statement generally starts with the Drake Equation looking at the number of observed stars, the number of stars with observed planets, the number of planets that could have earth like life (liquid water, enough light but not to much radiation) and guesses for how long life takes to develop -- and ends with the Fermi Paradox: "... but where is everybody?"
Simple life on earth appears to have developed reasonably quickly once conditions on proto earth allowed, and the sun is relatively young, which suggests that basic life probably has existed on many many planets since before our solar system existed.
A key point here is the difference between basic single cell life, tool using life(us) and life capable of crossing between stars or building structures we could see. Therefore current observations suggest that planets with slime molds are common, planets that have produced life capable of cheap faster than light travel is zero and planets with life somewhat like us is somewhere between the two. Many commentators are not explicit which case they are referring to.
In terms of remote detection of life, one plausible technique is looking at the spectra of planets looking for chemistry suggestive of life. This is beyond the resolution of current generation telescopes but is mathematically possible at least for nearby stars without implausible scaling up, and the questions one light year from earth is much closer than the nearest star (4.3 ly) so under ideal conditions (transit in front of the sun) current very large telescopes could probably identify earth as having liquid water, and probably the presence of on oxygen in the atmosphere, which would suggest some form of plant life going on. Detecting "tool-using life" is probably not possible at one light year unless something truly unnatural has been done to the atmosphere.