We use the tools we have available to make measurements of light and other particles that come our way and try to understand how the universe works under the assumption that things behave the same way here as "out there." That means that we assume that stuff out there is made of the same kind of stuff that is here on Earth and that the laws of physics here work the same everywhere else (and that we understand the laws of physics!).

So when we look at the light coming from a nearby star, we believe we know the properties of the star because we think we know how stars work, what they are made of, and how they should look. A question that you should ask yourself is: Are you satisfied by this? If not, what would it take to convince yourself that we understand what is out there?

In the springtime, how do you know what makes the distant mountains green?If you can't go there, what would you be able to tell from what is happening in your own neighborhood, right next to your house or school room? You might notice that after a rain, the grass grows green and the leaves come out on trees. From that you might guess that grass and trees are becoming green on the mountains as well.

The same principle is used when exploring deep space. We look at materials around us, and notice how they behave under different circumstances. When things get hot they glow in different colors depending on their temperature. When gasses are excited by light, they emit new light at particular wavelengths, forming lines in the spectrum. We see similar lines in the spectra of distant objects.
The lines behave in mathematically distinct ways, both here and in the distant object. Can we then say that the distant object has the same kind of gas that we have here in our laboratory? How many times will you watch cats drink water, before you say that all cats drink water?

There is another very important part to this answer:

We are honest with each other.

Each of us alone cannot possibly ask and answer all of the questions that help us understand what is going on in the distant Universe (or even at home!). We each rely heavily on what others tell us of their experience, in order to form our own understanding. We each make an honest effort to understand a new part of the "puzzle" that is our universe, then honestly tell that information to the rest of the community. This allows each of us to form a common opinion that has the greatest likelihood of being accurate and therefore useful.

The problems are already hard to figure out. If someone lies about their experience, then we all get confused and frustrated, and find it hard to believe anyone, even those telling the truth! If you knew someone whom you know lied about doing his or her homework, would you believe them if they said they studied the Moon and found water under its surface? Would you believe them enough to plan and take a trip to the Moon, not taking enough water because they told you there was enough already there?