How do scientists know that there are billions of other solar systems like us in the Universe?
They don’t know that, and I’ve never come across one that made that claim.
What they do know is that based on measurements of portions of the sky and extrapolation, that there are about 100 billion stars in our galaxy and up to 10 trillion galaxies in the universe. That means up to 1,000,000,000,000,000,000,000,000 stars. About 7.6% percent of those stars are class G stars (like our Sun).
We haven’t been to any of those stars to see if they have solar systems like ours. But we have been studying a very small portion of the sky in the constellation Cygnus, using the Kepler telescope. Kepler doesn’t allow us to visually see planets around far away stars, but it allows us to detect changes in the light coming from those stars caused by planets passing in front of the star. From this analysis, the estimates are now that almost all class G stars have at least one planet.
That means there are up to 76,000,000,000,000,000,000,000 stars similar to ours and almost all of them have some form of planets. Based on the Kepler observations, it is now estimated that a quarter of those stars have at least one rocky planet similar in size to the Earth and in the habitable zone.
That means there are up to 19,000,000,000,000,000,000,000 stars similar to ours with at least one planet similar to Earth.
All of this is based upon looking at small samples and extrapolating to the larger scale, but until data is found that indicates such extrapolation isn’t reasonable, we can — with some confidence — say that there are likely billions of other solar systems structurally somewhat like ours.
What we don’t have enough data to extrapolate is how many of those Earth-like planets have actually developed life. We have one data point thus far — Earth.