In the class, my professor gave the following example:
Let $f_n = \mathbf{1}_{[n,\infty)}$, then we have
$$\int_X \lim\inf f_n = 0, \text{ since } \lim\inf f_n = 0,$$ and $$\lim\inf \int_X f_n = \infty, \text{ since } \int f_n = \infty, \forall n.$$
I can understand "$\int f_n = \infty, \forall n$". But even so, what is the key reason that $$\int_X \lim\inf f_n = 0 \, \text{ but } \, \lim\inf \int_X f_n = \infty$$
To me for the second case, when $n\rightarrow \infty$, it should be $0$. I am confused on this point.
Well you are considering the sequence of functions $\{f_n\}$ where the $n^{th}$ function of the sequence is given by $$f_n(x):=\begin{cases}0 & \text{if } x\in (-\infty,n) \\ 1 & \text{otherwise} .\end{cases}$$
We have the pointwise limit $\lim_{n \to \infty} f_n(x)=0$, and so $\int \lim f_n =0$. But for each $n \in \mathbb{N}$ we have $\int f_n=\infty$, and so $\lim \int f_n=\infty$ (limit of a constant sequence).
The difference comes from what limit we are evaluating. On one hand we have the integral of a pointwise limit function, where the pointwise limit happens to be constantly $0$. And on the other hand we have the limit of a constant sequence of extended real numbers.