Context: LaTeX is a typesetting system. When compiling a document, a lot of really in-depth debugging information is printed, which can be borderline incomprehensible to anyone but LaTeX experts. It can also be a visual hindrance when looking for important information like errors.
The reason is that you're reading TeX, not LaTeX. The latter has abstracted away the fundamental building blocks so few people know how an hbox is set anymore. So, an hbox is a box where the content is in horizontal mode. Between the things is glue. Glue can stretch and shrink. Depending on how you have set your tolerance and penalties, there's a maximum percentage of stretch allowed. If the glue stretches more, it becomes bad, this is called badness and can effectively be up to 10000 bad. So why not just put more things into the box? Well, (La)TeX probably tried to do that, but came up with worse badness. TeX always chooses the least bad option on a paragraph level. In practice, the usual suspect is often that you have something else that can't fit the last part of a line, like a really long word. If you can look at it and manually hyphenate it, things might be better.
Thank you for this writeup, very informative. I get a lot of these "badness 10000" messages when working with things that have "complex" layouts, for example a resume/CV template. Given that TeX was originally made for research papers/articles, it makes sense that weirdness would arise when it's used for more layout-heavy stuff!
Badness 10000 usually indicates that something is very wrong. Usually overfull hboxes. If the text is spaced out to the point where it immediately looks bad, that could still be like badness 5000. What I have seen mostly is macros not playing well with other macros, and in LaTeX there's a lot of macros under the hood, so it's very hard to troubleshoot.