Other answers are great - this may fill in any gaps. I'm not going to talk about what should be, just what is (or was before COVID). I will mention there are arguments against conferences that you don't bring up, especially inequity across geographic locations. I will also mention that CS has a fair number of great journals in addition to conferences and one could publish mostly in these if one wanted to.
But does the word conference here carry the same meaning as it does in other fields?
No - I think the event is similar, but in some fields there is little reviewing process and one may speak about the same work at multiple conferences, whereas at main CS conferences the reviewing process is relatively rigorous and accepted papers are widely consider "published", and only appear at one conference. However:
If your paper is accepted in a conference in computer science, does it simply mean that it will be published in a proceedings rather than a journal?
Usually yes, but it is also common to publish a more extensive or detailed version of the same work in a journal later -- some journals and conferences have explicit policies and agreements about this.
Or does it really mean that you have to travel to a physical conference and present your work as a talk?
As others mentioned, it's expected that you only submit if one of the authors plans to go and present if accepted. In unusual circumstances like denied visas it can be worked around.
If the latter, what happens if you have multiple papers accepted at different conferences in the same year but can't afford to travel to all of them?
This seems rare for several reasons. First, you plan travel when planning submissions, so you just wouldn't submit to those if you didn't have budget.
But mostly, almost everyone who publishes at these conferences is a professional researcher or close enough that their employer subsidizes/funds their travel. Tech companies pay for publishing employees to go (usually) and academics use research funds. This is part of their yearly budget. Many/most presenters are grad students who get somewhat reduced rates and often can get travel grants from different sources.
Also, most computer scientists collaborate with one to four or more other people on most papers, so each individual has flexibility. Often the least-senior author is given the opportunity to present and has funds available, so more-senior authors often don't travel, or travel to participate in the conference but don't present.
I imagine that in this situation, trying to publish any work would result in a scheduling nightmare, both on the part of conference attendees and organisers.
Again most participants are full-time researchers so they plan on attending the same conferences, or a subset of the same, every year, often regardless of whether they happen to have a paper accepted or not. Each conference is usually held around the same time each year so they don't usually conflict with other conferences in the same research area.
If this is the case, how did this situation arise and why is it allowed to continue?
I don't know much about how it arose, but why does it continue - it works well for a lot of people, especially people in power of decisionmaking. The case hasn't really been made for compelling alternatives.