How do survival models "account for censoring"? (Do they?) Background
I'm teaching an intro stats class in our social / health sciences department and I'm finding myself tripped up on something I'd always taken for granted: namely, the claim that survival analysis methods (from Kaplan-Meier to Cox models) "account for" censored data, and that this is one of these methods' central advantages over other approaches.
(Relevant background: in my research I use, mostly clumsily, some applied statistical and quantitative methods, and I have a good grasp of and intuition for some of the basics. So I'm an okay generalist but I'm not a statistician nor even a statistics grad student by any means.)
The Problem
As I'm preparing some PowerPoint slides for these students, though, I'm realizing that while I have a pretty good grasp on what censoring is in survival methods, I have no idea how something like a Cox model "accounts" for it. I'd never thought about it, and instead assumed that this is just what happened. (It's possible that a professor walked us through this earlier in my education, but it's also possible that I used to struggle in 8am stats lectures.)
Censoring, as you all already know, is the state in which we have some information about individual survival time, but we don't know the survival time exactly (Kleinbaum & Klein, p. 5; I've always liked this plain English explanation). In my experience, sometimes when stats people are trying to sell someone on survival methods, they'll say things like "logistic regression doesn't account for censoring, but Cox regression does!"
There are some great posts on CV about censoring; this is maybe my favorite example. In it, user @Tim gives a terrific explanation of censoring:

Intuitive example of censoring is that you ask your respondents about their age, but record it only up to some value and all the ages above this value, say 60 years, are recorded as "60+". This leads to having precise information for non-censored values and no information about censored values.

I think this is brilliant (in fact I plan on borrowing liberally from it during the lecture, with credit given of course). But it doesn't get really get into how survival analyses actually deal with this, and whether it's a selling point for survival methods ("our methods can do this and yours do not") or just something that pops up when you try and ask survival-type questions ("how long will people live, on average, after being given treatment X?").
The Questions

*

*Is it true, strictly speaking, that something like a Kaplan-Meier estimator or a Cox proportional hazards model is "accounting" for censoring?


*If so, how is it doing that?


*If a survival model indeed accounts for censoring, is this a "feature" of survival methods over others, or a "bug", an inevitable artifact of the sorts of questions one uses survival methods to answer?
My guesses
Well, not guesses, maybe more like very unclear intuitions I'm not too confident about:


*Why is censoring a problem? I'm thinking that, if ignored, censoring is a huge potential source of bias in making survival estimates: if you don't know what happened after Mr. Smith dropped out of your study (i.e. was lost to follow-up, i.e. was censored) your estimates may be off in one direction or the other. Maybe he lived a long, long time -- or maybe he died the next day. If this is happening to lots of people in the same way, your estimates may be really, really off.


*So maybe what survival models have found a way to do is keep in the analysis everyone who contributed survival time, regardless of whether we know about their outcome status, while other methods would simply drop all of those people's data as missing.
Am I way, way off here?

Update: @James Stanley address the same point in his answer; see point #4.

This is not an answer but an extended comment to clarify some terminology and highlight an important assumption.
You write:

if you don't know what happened after Mr. Smith dropped out of your study (i.e. was lost to follow-up, i.e. was censored) ...

implying that "loss to follow-up" is the same as "censoring". Not exactly.
Say we do a randomized control trial (RCT) of treatment T vs control C and the outcome of interest Y is the time to event (or equivalently, survival). We start recruiting subjects on January 1st and the study concludes on December 31, the same year. We follow up with subjects every month from recruitment until Dec 31.

*

*loss to follow-up: The patient A is recruited and randomized to T or C in Jan. We have data for the first 8 visits and no data for the last 3 visits from Oct to Dec. We know the patient survived for at least 8 months.

*censoring: The patient B is recruited and randomized to T or C in Apr. We have data for all 8 visits until the study concludes in Dec. We know that the patient survived for at least 8 months.

You may ask: Why make the distinction between loss to follow-up and censoring? Patients A and B appear "similar" ("exchangeable"): we followed them for 8 visits post-treatment, so we know they survived at least 8 months, but we don't know how long they survived.
They would contribute the same term $\operatorname{S}(t_i = \text{8 months} | \theta)$ in the likelihood function. [Here I refer to the likelihood $L_t(\theta)$ in @Ben's answer.]
The subtlety is that we don't know why patient A dropped out from the study. Often in analysis when we don't know something, we assume that it happened randomly. This will make the math work out just as described in Ben's answer.
However, suppose that the treatment has severe side effects and patients who received the treatment have a higher probability to discontinue treatment early. This means that subjects in the treament group are more likely to be lost to follow-up. This (potentially) introduces bias in the estimate of the treatment effect, unless more sophisticated analyses come to the rescue.
