So having for example $f:\Lambda^3_2 \to X$ is having the data of
- 4 0-simplicies $x_0,x_1,x_2,x_3$,
- 6 1-simplicies $f_{0,1},f_{1,2},f_{2,3},f_{0,2},f_{0,3},f_{1,3}$ that we can interpret as 'morphisms' $f_{i,j}: x_i \to x_j$ for $i<j$ in $\{0<1<2<3\}$, meaning the two 0-faces of $f_{i,j}$ are $x_i$ and $x_j$,
- 3 2-simplicies $f_{0,2,3}, f_{0,1,2}, f_{1,2,3}$, ($f_{0,1,3}$ beeing the one missing), that we can interpret as homotopies witnessing compositions : for example $f_{0,2,3} : f_{2,3}\circ f_{0,2} \sim f_{0,3}$,
- no 3-simplex.
So the horn filling condition states that given that data you can find the missing 2-simplex $f_{0,1,3}$, and the missing 3-simplex $f_{0,1,2,3}$, this 3-simplex, i.e. a homotopy of compositions of homotopies : $$f_{0,1,2,3} : f_{0,2,3} \circ f_{0,1,2} \sim f_{0,1,3} \circ f_{1,2,3}$$
The first composition of homotopies, $f_{0,2,3} \circ f_{0,1,2}$, says that we can decompose $f_{0,3}$ as $f_{2,3} \circ f_{0,2}$ first and then decomposite further to get $f_{2,3} \circ (f_{1,2} \circ f_{0,1})$.
The second composition of homotopies, $f_{0,1,3} \circ f_{1,2,3}$ says we can decompose $f_{0,3}$ as $f_{1,3} \circ f_{0,1}$ first and then as $(f_{2,3} \circ f_{1,3}) \circ f_{0,1}$.
So this 3-simplex encodes assosciativiy of the compositions of 3 morphisms up to homotopy, a homotopy between $f_{2,3} \circ (f_{1,2} \circ f_{0,1})$ and $(f_{2,3} \circ f_{1,3}) \circ f_{0,1}$.
I like to see 1-simplicies $f_{i,i+1}$ as the morphisms I want to compose, and $f_{i,j}$ with $|i-j| > 1$ as a candidate for compositions, so here for example let's say $f_{0,1} = f$, $f_{1,2} = g$, $f_{1,3} = h$, then $f_{0,2}$ will be a candidate for the composition $g \circ f$, $f_{1,3}$ will be a candidate of the composition $h \circ g$, and $f_{0,3}$ a candidate for composition $h\circ g \circ f$, but this $f_{0,3}$ can a candidate for $h \circ g \circ f$ in two ways : $f_{1,3} \circ f_{0,1}$ i.e. $(h\circ g) \circ f$ or $f_{2,3} \circ f_{0,2}$ i.e. $h\circ (g \circ f)$. 2 simplices when they exist are ways to compose and 3 simplicies are ways to compose those compositions and so on.
4-horns can be interpreted in a similar way but it is tedious to write down and you need to have good vision in 4-dimensions.
If you write down some small simplicial set with an easy geometric description (e.g., the simplicial set corresponding to some simplicial complex), it will almost never be a Kan complex. This should be evident if you get some practice working with Kan complexes and using the horn filling condition: it implies your simplicial set must be very "rich" in simplices and that you can "combine" simplices in it similar to the way you can combine singular simplices in a topological space. This is highly incompatible with how simplices behave in a typical nice triangulation of, say, a manifold.
For a really simple example, consider the $1$-simplex $\Delta^1$. Note that there is a horn $\Lambda^2_0\to\Delta^1$ given by mapping the vertices $0,1,2$ to $0,1,0$, respectively ($\Lambda^2_0$ only has the edges $0,1$ and $0,2$, and this map sends them the the edge $0,1$ and the degenerate edge $0,0$). This horn cannot be filled, since there is no edge from $1$ to $0$ in $\Delta^1$.
This example more generally shows that in any Kan complex, edges must be "reversible": if there is an edge $e$ beween two vertices, there is another edge $\bar{e}$ between the same vertices in the opposite order, such that $e$ and $\bar{e}$ together form a 2-simplex where the third edge is degenerate. This property can never hold in a $\Delta$-complex with any edges, since if $e$ was nondegenerate, the 2-simplex will also be nondegenerate, but a nondegenerate simplex in a $\Delta$-complex cannot have a degenerate face. So no non-discrete $\Delta$-complex is a Kan complex. On the other hand, this property is familiar from the context of arbitrary continuous paths in a topological space, which can be reversed.
Best Answer
Just by googling:
https://arxiv.org/abs/2005.04853 Cubical models of (∞,1)-categories.
(The authors use additional structure ["connections" or "markings"] to obtain the desired model structure).