For instance, what does it mean to say that the lower limit topology on $\mathbb{R}$ is strictly finer than the usual topology on $\mathbb{R}$?
I understand why lower limit topology is finer.
Take any open $(a,b), b>a$ in $\mathbb{R}_{usual}$, we can represent $(a,b) = \bigcup\limits_{n \in \mathbb{N}} [a+\dfrac{\epsilon}{n}, b)$, where $\epsilon < \dfrac{b-a}{2}$
Then all the sets in $\tau_{usual} \subseteq \tau_{ll}$.
What does it mean for the lower limit topology to be strictly finer?
Are there two topologies such that one is finer but not strictly finer?
Summarising the comments:
The lower limit topology being strictly finer than $\tau_{\text{usual}}$ means it's not equal to $\tau_{\text{usual}}$.
It's essentially the difference between $<$ and $\le$: $x < y$ means exactly $x \le y$ and $x \neq y$. Or really, the difference between $\tau_1 \subsetneq \tau_2$ and $\tau_1 \subseteq \tau_2$.
The only way $\tau_1$ can be finer than $\tau_2$ but not strictly finer, is when $\tau_1 = \tau_2$.
It follows also that these two topologies cannot be strictly finer than each other, just like we cannot have $x < y$ and $x > y$ at the same time.
If $\tau_1$ is finer than $\tau_2$ and $\tau_2$ is finer than $\tau_1$ (no stricter), then we have $\tau_1 \subseteq \tau_2$ and $\tau_2 \subseteq \tau_1$, so $\tau_1 = \tau_2$.
So to see the strictness in your example, it suffices to notice that $[0,1)$ is open in the lower limit topology, but not in the usual topology, e.g.