Intuitive meaning of immersion and submersion

First of all, note that if $f : M \to N$ is a submersion, then $\dim M \geq \dim N$, and if $f$ is an immersion, $\dim M \leq \dim N$.

The Rank Theorem may provide some insight into these concepts. The following statement of the theorem is taken from Lee's Introduction to Smooth Manifolds (second edition); see Theorem $4.12$.

Suppose $M$ and $N$ are smooth manifolds of dimensions $m$ and $n$, respectively, and $F : M \to N$ is a smooth map with constant rank $r$. For each $p \in M$ there exist smooth charts $(U, \varphi)$ for $M$ centered at $p$ and $(V, \psi)$ for $N$ centered at $F(p)$ such that $F(U) \subseteq V$, in which $F$ has a coordinate representation of the form $$\hat{F}(x^1, \dots, x^r, x^{r+1}, \dots, x^m) = (x^1, \dots, x^r, 0, \dots, 0).$$ In particular, if $F$ is a smooth submersion, this becomes $$\hat{F}(x^1, \dots, x^n, x^{n+1}, \dots, x^m) = (x^1, \dots, x^n),$$ and if $F$ is a smooth immersion, it is $$\hat{F}(x^1, \dots, x^m) = (x^1, \dots, x^m, 0, \dots, 0).$$

So a submersion locally looks like a projection $\mathbb{R}^n\times\mathbb{R}^{m-n} \to \mathbb{R}^n$, while an immersion locally looks like an inclusion $\mathbb{R}^m \to \mathbb{R}^m\times\mathbb{R}^{n-m}$.