Trying to understand what Travis CI does and when it should be used

The simplest way to explain Travis CI is that it runs your program's tests every time you commit to GitHub (this can be configured in many ways, and you can always disable builds on some branches). The point of this is that you can often discover very quickly if your commit broke something, and fix it before it becomes a problem. I would recommend running Travis CI on every GitHub repo that you have unit tests in and is using a programming language supported by Travis CI. Since setting up Travis CI is very easy, I don't normally see a good reason not to use it, unless you don't care if you have passing tests in your program or not. Feel free to leave a comment if you have any more questions. You can read more about Travis CI here.

As you have already discovered what is Travis-CI, I would directly point to the questions you have.

By building the project and run tests, what tests is it going to run? And how is it going to "build" the project? (like compiling it to binary?)

In the .travis.yml a file you're specifying your OS, the programming language, your repo branch, the project file name and other details. By reading this file, Travis-CI will use the specific compilers which installed on their server to compile our code. Probably they will have the same mechanism as we have for Github. For the first time, they might pull the code [if we have specified specific branches they might pull the code from those branches only]. Also, we have authenticated to use our account with Travis-CI, whenever we make a commit, there should be some notification should fire to Travis-CI server thus it will be recognized as a commit and it will start compiling.

It states that "This includes commits to all branches" - but what if I don't want to commit to all branches?

You can specify different branches or the master branch. And it should only compile the specific branches specified in .travis.yml file.

Is it alright if I don't use Travis Cl at all? Under what circumstances is it best to use it (or it must be used)?

Yes, it's alright. Not a big deal. But what benefits you will be missing by not using this easy to integrate engine with your repo. Everytime you commit it may possible that it miss something and it couldn't compile because of a code. How will you know? Thus, you should use Travis-CI.

_{I have written a blog post which you can read to know, what is Travis-CI, Continuous Integrations and how to linked Travis-CI with your Github Repo. I have written it for a Swift repository.}

I think am in a very good position to answer you question as am currently learning Travis CI at the moment. The first problem you encountered in the first place is because the project you are contributing to is using Travis CI to test and build the project. If the project isn't using Travis CI, you wouldn't have come across such error.

The solution is checkout Travis CI website and learn how it work and how it affected the project you are working on. This will put you in a very good position to understand what's failing in your code and how to fix it.

To answer the first question about building the project.

By building the project and run tests, what tests is it going to run? And how is it going to "build" the project? (like compiling it to binary?)

It's means how the source code is processed before use/test. It depends on the language you are writing. For example, if the project is writing with PHP. It's not going to build my code into executable file like C/C++ source code. It will run my PHP code on PHP interpreter and test it as it does so. It still goes through the normal compiling process your preferred language goes through.