The way I think of ancillary statistics is: a data set has numerous sources of randomness. A sufficient statistic carries all the information that I can extract in the data about this randomness. A minimally sufficient statistic is the "smallest" sufficient statistic. Now say that this has the form $S = (T,C)$.
Now I happen to be interested only in some aspects of the randomness ($T$) in the data, and I would rather condition on the uninteresting parts (treat $C$ as actually fixed to $c$). This makes sense to do when $C$ depends only on parameters that I am not interested in, and it is especially useful when $T|C$ has a simple distribution.
For example, I carry out a linear regression. I am interested in how some covariate affects the outcome, and I assume that $Y_i \sim N(\alpha + \beta x_i, \sigma^2)$ for $1 \leq i \leq m$. What I care about is $\alpha, \beta$ and $\sigma^2$.
But there are also other sources of randomness: $x$ is probably realization of a random variable $X$ (uninteresting). The sample size is probably random as well (if it is not fixed in advance), and $m$ is a realization of a random variable $M$ (uninteresting). When I carry out my linear regression I condition on these aspects of the data. One reason is that I am not really interested in $X$ and $M$ per se. The second reason is that it would be very complicated to write a likelihood for all sources of randomness $(Y_i, X_i, M)$, $1\leq i\leq M$ at the same time.
If there is a minimal sufficient statistic $S=(T, C)$, and $C$ describes exclusively $X$'s and $M$ then I would rather condition on $C=c$, which is the same as treating $X_i = x_i$ and $M = m$ as fixed.