It turns out that when light travels through the mostly emptiness of space, it can travel amazingly far without being significantly diminished. So we see galaxies the same way we see stars - you can see some with the naked eye, in fact, but to see detail you need a telescope. They come in somewhat different configurations, but generally look like this:

We don't know exactly how many there are, we have to estimate it, and the range of estimates is huge: between 100 billion and 200 billion!

How do we estimate it? By counting the galaxies in a teeny tiny area of the sky, and then multiplying by that fraction. There are an amazing number of galaxies that are viewable in an area of the sky the size of the head of a pin held at arm's length. It looks like this;

By the way, the galaxy at the top of this comment is eighty million light years away. That means the light were seeing from it now left eighty million years ago; we're seeing back in time.

have you seen the Hubble Deep Field image?

it's a picture of a section of the sky "about the size of a tennis ball at a hundred meters" taken by pointing the telescope at a point that appeared completely dark over a period of 10 days. there are over 3000 galaxies in that image.

We make the following assumptions:

1. We are not in a privileged or special place in the universe.
   1a. Therefore the universe should look the same from where we are as from almost anywhere else.
2. Supernovae emit a very reliable amount of light.
   2a. Therefore we can accurately tell how far away a supernova is by its light signature.
3. We can tell how far away distant galaxies are when a supernova goes off in them.

Therefore, we can map out and count the galaxies in our local area, and, using assumption 1, we know that the number of galaxies in our volume of space is the same as in every volume of space. Multiply the one by the other and you've got your answer.

Here's a very zoomable image of the sky. If you zoom in, you can see galaxies. If you then zoom out a bit you get a sense of how many there are.

https://www.legacysurvey.org/viewer#NGC%206339

Just look at this picture: https://esahubble.org/images/heic0406a/
This is my go to when my problems start to feel too big :D

Didn't you know that Stephen Hawking has counted them all?

/s

By pointing a telescope at them and looking. You can see some of the larger ones through a cheap home telescope for a couple hundred dollars.