The well-known, well-tested and well-understood laws of physics can be used to take the universe in its present state and answer the question: what did the universe have to have looked like 1 million, ten million, one hundred million, one billion, and finally 10 billion years ago in order for it to have evolved into exactly what we see today?
Those laws place extremely stringent constraints upon what the so-called "initial conditions" of the universe could possibly have been in order to eventually furnish us with today's universe- and predict for us what evidence if any was left behind at each epoch in the universe's evolution through time. The best example of this is the Cosmic Microwave Background, which is a relic- a fossil- left behind after the Big Bang had gone to completion and launched the universe onto its evolutionary course. This fossil radiation was predicted to exist years before the instruments that eventually found it (see below) had been built.
Then we train our astronomical instruments upon the heavens and search for that evidence. Note that without those instruments, we humans could not possibly see those signals because they are far too faint for our eyes, or they consist of signals that our eyes are not capable of detecting (radio waves, infrared, ultraviolet, or x-ray light, etc.). Remember: just because out eyes can't detect radio waves does not mean that they are not "real".
Note also that when we look at far-away objects, we are seeing them not as they are today but as they were hundreds of millions or even billions of years ago when the light we see today first departed those objects. This means that a powerful telescope is in fact a time machine which lets us look into the past and see just what the universe was like then.
Dozens of excellent books have been written on this topic by the experts in the field. My favorite is The First Three Minutes by Stephen Weinberg.