Reading about the history of settlements in the western USA, they often mention the establishment or departure of a Post Office. What was the significance of those?

Without a post office, it was much harder to communicate with the outside world, since you had to have your mail delivered to a village or town some distance away. At the time, there was essentially no private parcel or letter delivery services outside the major cities.

We don't realize it as much in the 21st century, but before email and telephones, if you wanted to communicate with someone that wasn't next to you, you either need to travel to see them in person, or write a physical snail-mail letter an send it.

A town without a post office would have a harder time attracting new residents and new businesses. The absence of a post office made it harder to send both goods and information.

Before 1896 there was no rural free delivery (to a household). If you lived outside of a major city, you needed to go to the local post office in person to get your mail -- the local office would collect it as it came in, and hold onto it until you came into town to pick it up, whenever that was. In fact, before 1863, there was no free city delivery, either.