According to the definition of sufficiency, a statistic is sufficient for a parameter if the conditional distribution of $X$ given a value of statistic does not depend upon the parameter.
What I am trying to understand is how does conditional distribution of $X$ not being a function of the parameter fit into the intuitive meaning of sufficiency i.e statistic value holding the same amount of information as that of the respective sample.
If the distribution of something you observe does not depend on a parameter, it cannot possibly give you information about it.
Now, if the distribution of $X$ depends on the parameter $\theta$ and the distribution of $X$ given the sufficient statistic $S$ does not, it must be the case that all information about $\theta$ is in $S$; once the value of $S$ is given, the value of $X$ becomes irrelevant, because the conditional distribution of $X$ no longer depends on $\theta$
We need some more notation. Suppose the random variable $X$ has a distribution from some family $f(x; \theta)$ parametrized by $\theta \in \Theta$. Suppose that $T=T(X)$ is a sufficient statistic (for $\theta$.) Then by the factorization theorem we have $$ f(x; \theta)= h(x) g(T(x); \theta) $$ where $h$ is a function not depending on $\theta$. Now, using the result from Can the Fisher factorization theorem be understood as a product of densities?, this can be interpreted as a factorization of the distribution of $X$, and we can use this to simulate from the distribution of $X$ by first simulating $T$ and then simulating from the distribution of $X \mid T=t$.
So after having observed $T(x)=t$, we can simulate surrogate data having the same distribution as $X$ by simulating from the above conditional distribution, which by sufficency do not depend on $\theta$. This is a way of giving intuitive meaning to sufficiency; knowing only $T(X)=t$ we can recreate by simulation surrogate data having the same distribution as $X$.
There are other ways to get intuitive meaning to $T$ having the same information content as $X$, via its use in inference. Without going into details
The mle (maximum likelihood estimator) of $\theta$ is a function of $T$ (or if nonunique, can be chosen in such way)
given a prior for $\theta$, the bayesian posterior will be a function of $T$
and there are many more general results of this sort. The two above should be easy exercises.
