What is independent censoring and what are its assumptions? I'm reading the Introduction to Statistical Learning book. In chapter 11 (Survival Analysis, page 463), the authors state:

In general, we need to assume that the censoring mechanism is
independent: conditional on the features, the event time T is
independent of the censoring time C.

I also read the given two examples:

In order to analyze survival data, we need to make some assumptions
about why censoring has occurred. For instance, suppose that a number
of patients drop out of a cancer study early because they are very
sick. An analysis that does not take into consideration the reason why
the patients dropped out will likely overestimate the true average
survival time. Similarly, suppose that males who are very sick are
more likely to drop out of the study than females who are very sick.
Then a comparison of male and female survival times may wrongly
suggest that males survive longer than females.

But I don't think I fully comprehend it (because I'm not confident while doing the exercises). Could you explain to me what independent censoring is in another way and give me some examples?

$\rm [I]$ provides a nice intuitive discussion on independent censoring; its assumption and necessity.
Suppose there are two groups: $\rm A, ~B,$ each group consisting of $100$ subjects initially disese-free.
Below is the schematics of $\rm A:$
$$\underline{\text{Group A}}\\\begin{array}{|c|c|c|c|}
\hline
\text{Time}   & \text{At risk} & \text{Events} & \text{Survived}              \\ \hline
{0-3}    & {100}     & {20}     & 80                    \\ \hline\end{array}\\\text{40 censored}\\
\begin{array}{|c|c|c|c|}
\hline
\text{Time}   & \text{At risk} & \text{Events} & \text{Survived}              \\ \hline
{3-5}    & {40}     & {5}     & 35                    \\ \hline\end{array}$$
For $\rm B:$
$$\underline{\text{Group B}}\\\begin{array}{|c|c|c|c|}
\hline
\text{Time}   & \text{At risk} & \text{Events} & \text{Survived}              \\ \hline
{0-3}    & {100}     & {40}     & 60                    \\ \hline\end{array}\\\text{10 censored}\\
\begin{array}{|c|c|c|c|}
\hline
\text{Time}   & \text{At risk} & \text{Events} & \text{Survived}              \\ \hline
{3-5}    & {50}     & {10}     & 40                    \\ \hline\end{array}$$
How would one estimate $5$-years survival for each group?
Assume censoring is independent that is, it is random
within any groups $\rm A, ~B.$
So under the assumption of independent and random
censoring, in $\rm A,$ it can be expected that the $40$ subjects who were censored were similar to the $40$ who remained at risk with respect to their survival experience. Since $5$ out of those $40$ were remained under observation experienced the event, then $5$ would be the estimated number out of those censored $40$ over that same time span. So, estimated $5$-years survival for $\rm A$ would be $\frac{20 +5 + \underbrace{5}_\text{from censored cases}}{100} = 0.70.$
In the same vein, one can calculate for $\rm B,$ the estimated $5$-years survival which would be $0.48.$
The reason for stating two groups is to discern between independent and random censoring: observe $\rm A$ has higher proportion of censoring ($0.50$) than that of $\rm B$ ($0.17$). The censoring is not random. However, conditional on group status, the censoring is random. This is the basis of independent censoring.
Formally, independent censoring means

within any subgroup of interest, the subjects
who are censored at time $t$ should be representative of all the subjects in that subgroup who
remained at risk at time $t$ with respect to their
survival experience.

The authors aptly summarized the assumption:

Assume survival experience of subjects censored at $t$ is as expected if
randomly selected from subjects
who are at risk at $t$.

That is,

Even though the censored subjects
were not randomly selected, their survival
experience would be expected to be the same
as if they had been randomly selected from the
risk set at time $t$.


Reference:
$\rm [I]$ Survival Analysis: A Self‐Learning Text, David G. Kleinbaum, Mitchel Klein,  Springer Science$+$Business Media, $2012,$ presentation $\rm XI.,$ pp. $37-41.$
