Like everything in mathematics, "obvious" is a term of art. Broadly speaking, it refers to a fact, proof, consequence, which is necessary for the proof to advance, but which is already established elsewhere, so it does not in itself aid in understanding the proof being presented.
A proof is either providing a new result, or is proving an established result in a new way. Almost always, a proof will need other results, in a way that isn't "interesting" (another term of art). The point of introducing these results as "obvious" is basically to say "here is something which isn't proven by the proof I'm presenting, we need it, but there's no need to derive it to follow this proof", ideally, with a footnote. As language, it is a bit sly: if something is obvious in the normal sense, it will be left out.
It's a problem that the modern style is to elide anything obvious in this sense, rather than in the sense of "anyone who might reasonably read this paper may be expected to know this". But labeling these things as obvious isn't meant in the sense "if you don't know this, you're stupid or uninformed", or in fact "this will be instantly clear as soon as I mention it", it's meant to mean "if you were to follow the breadcrumbs and check up on the 'obvious' thing, it wouldn't help you much in following my proof, so take my word for it. Or, y'know, knock yourself out if this step is interesting to you".