How to get this library to work load?

Ask your hosting service to install the library for you.  My host does this for me all the time -- all it takes is a phone call!

I asked and they said no - it's shared hosting. Unfortunately I'm stuck with the corporately approved hosting provider.

How can I load this library without Composer?

I saw your Gig.  You don't need to use composer.  Just download the the git and upload to your hosting service.  

There are some libraries that are difficult to run without composer, this does not appear to be one of those.

Without composer, it looks like you would change the settings.php file to your own and include that https://github.com/vgrem/phpSPO/blob/master/examples/Settings.php

The Settings is just an array of variables. It doesn't load anything.

I already tried downloading, uploading to my server... I don't really know after that how to load it.

The vendor/autoload.php wants composer.

Note the examples all use autoload.php as well.

I can look at this tomorrow.   Another option I have done in the past is to run composer locally with the same file structure as my web server, then upload everything.  You don't have to have composer running on the web server. But the autoload created locally saves a lot of work.

This worked for me as well - I can connect and read the SP site. Though i have an issue adding items, I'll post another Q. So thanks.