Converting vectors to tangent or world space. (Explanation please?)

Hi, in the tutorial I’m watching, the Vector Transform node is used to change a vector from tangent to world space. I’ve taken a look at the documentation for this “Transform” node:

“The Transform expression converts a three-channel vector value from one reference coordinate system to another.
By default, all shader calculations in a material are done in tangent space. The vector constants, camera vector, light vector, etc are all transformed to tangent space before being used in a material. The Transform expression allows these vectors to be transformed from tangent space to world-space, local-space, or view-space coordinate systems. In addition, it allows world-space and local-space vectors to be transformed to any of the other reference coordinate systems.”

Here’s the example picture that can be found on the wiki page.

Can anyone walk me through exactly what is going on here? I understand that tangent space is “up and down” while world space can be in any number of directions, but I don’t understand how it could be applied to something like a flat material. And I don’t understand how you could just take an arbitrary constant vector and convert it to world space- it seems like just a number, it doesn’t actually /exist/ anywhere. Are the “tangent / world” spaces based on the flat material themselves or the object instance they’re applied to?

Maybe someone could carefully walk me through exactly what is happening in this example. I don’t have a lot of experience with higher-level math terms so forgive me.