In simple terms “Simulate” means “Mimicry/Copy or imitate something”.
I blink an eye, you blink an eye. I walk, you walk. I run, you run.
Basically, You are simulating me. You are simulating “my behavior”.
If you can simulate each of my behavior/activity, then you are at least as powerful as me(You maybe more powerful).
But If I also can simulate each of your behavior/activity, then we are equally powerful.
For Example : Jio Chat simulates design of Whatsapp.
https://www.mensxp.com/technology/news/78092-jiochat-looks-like-a-copy-of-whatsapp-we-wonder-if-jio-spends-any-of-its-investment-on-design.html
Now, What behavior can DFA, NFA etc simulate(mimic/imitate) of each other? What do we mean by their “behavior” ?
DFAs and NFAs are computational models that solve decision problems. By their “behavior”, we mean the language they recognize.
So, When we say that “an NFA can simulate a DFA”, we mean that if we have a DFA $D$ whose language(behavior) is $L(D)$ then we can have some NFA who also has same behavior(Language).
When we say that “a DFA can simulate an NFA”, we mean that if we have an NFA $N$ whose language(behavior) is $L(N)$ then we can have some DFA who also has same behavior(Language).
So, let’s ask two questions:
$\color{red}{\text{Can DFA be simulated with an NFA ?}}$
Answer : Obviously. Every DFA is a special kind of NFA. Thus if a language $L$ is recognized by a DFA $D$, then since $D$ is a special NFA, $L$ is also recognized by an NFA.
$\color{red}{\text{Can NFA be simulated with a DFA ?}}$
Yes. We can construct a deterministic finite automaton (DFA) to simulate the NFA $N$. For any NFA $N$, We can use the “set-of-states construction/Subset Construction/Powerset Construction” to get a DFA whose language is same as the NFA $N$.
https://courses.engr.illinois.edu/cs374/fa2018/notes/models/NFAnotes.pdf
https://people.csail.mit.edu/rrw/6.045-2020/lec2-color.pdf