Why is spacetime in special relativity hyperbolic? In Galilean relativity, spacetime is 4 dimensional, flat and Euclidean.
In Einstein's (special) relativity spacetime is 4 dimensional, flat but hyperbolic.
Why is it hyperbolic?
 A: No, spacetime is not 4 dimensional Euclidean in Galilean relativity; it's 3+1 dimensional, where the time dimension is completely separate from the spatial dimensions and does not mix with them at all.
In special relativity time does mix with space, to a degree. But it's an empirical fact that time and space are not the same thing. Thus the 1 temporal and 3 spatial dimensions do have some difference, and in the Minkowski metric this turns up with them having different signs (the line element is $ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$), which makes spacetime hyperbolic.
So the bottom line is... that's just the way the world is. If space and time were truly the same we'd live in a 4d Euclidean universe, but then we could travel in time in exactly the same way that we can travel in space, and it's not clear in that circumstance what "time" would even mean. Certainly the universe would be very different indeed.
A: related: Is spacetime in special relativity hyperbolic?
ANS: The spacetime in special relativity is not hyperbolic (is not a [curved] hyperbolic geometry).

Similarly, the geometry of Euclidean space is not "circular" or "spherical"
(is not a [curved] spherical geometry).

The spacetimes of Galilean relativity and Special Relativity are special cases of the Cayley-Klein geometries. Both are flat (zero curvature) geometries, like Euclidean space.

So, the parallel postulate applies.
(Because of this, we can do vector algebra!)
The "circles" (or generally "hyperspheres") of constant square-interval are the following 3-dimensional hypersurfaces:

*

*circles (generally "hyperspheres") in the Euclidean case.
Note: In 2-dimensions, we use the "circle" for "circular trigonometry" in this flat [zero-curvature] 2-dimensional space.


*hyperbolas (general "hyperboloids") in the special relativistic case.

Note: In (1+1)-dimensions, we use this "hyperbola" for "hyperbolic trigonometry" in special relativity in this flat [zero curvature] 2-dimensional space (Minkowski spacetime).

This is not the same as the curved "hyperbolic geometry" of Lobachevsky.


*horizontal lines (generally horizontal "hyperplanes", assuming time upwards upawrds) in the Galilean case [as in the PHY 101 position-vs-time graphs].
In relation to "maximum signal speeds"

*

*In special relativity, the "speed of light" (i.e., the maximum signal speed)
is finite and corresponds to the light-cone, a hyperboloid of radius 0. The light cone is asymptotic to the hyperboloids with nonzero square-radius. Thus, the reason for the hyperboloids is the speed-of-light-principle together with the relativity-principle.


*In Galilean relativity, the maximum signal speed is infinite.
The horizontal hyperplane corresponds to taking the limit of "opening up the light cone" for an infinite maximum signal-speed.
Comments on the hypersurfaces and their curvature:

*

*The tangent hyperplanes to the "hypersphere" are 3-dimensional and define the notion of perpendicularity to the radius vector to that point of tangency. The hyperplanes have a Euclidean geometry.


*In the Euclidean case, the tangent hyperplanes to a given hypersphere are distinct.
Similarly, in the special relativity case, the tangent hyperplanes to a given hyperboloid are distinct. This is why simultaneity is relative [not absolute].
The hypersphere and the hyperboloid are 3-dimensional hypersurfaces with nonzero constant curvature, and thus have a non-Euclidean geometry: Riemann's elliptic geometry and Lobachevski's hyperbolic geometry.


*Since the Galilean-sphere is itself a hyperplane, all tangent hyperplanes
coincide with the Galilean-sphere itself. (This is why simultaneity is absolute in Galilean relativity.) So, the Galilean-sphere (a 3-dimensional hypersurface) also has a Euclidean geometry.

In short, in special relativity,
using a "hyperbola" for a "circle" (curve of constant square-interval) makes the trigonometry "hyperbolic trigonometry" in a flat space
(but this does not make the geometry a "[curved] hyperbolic geometry",
where the parallel postulate fails).
A: It's an outcome of the absolute nature of the motion of light. But to begin at the beginning ...
Since antiquity, the notion of rest and motion were distinct ideas. One was the negation of the other. Galilean relativity unified them and said rest was the same as uniform linear motion. In this geometry space is Euclidean.
Now, after Maxwell discovered his equations for electromagnetism, he identified light with electromagnetic radiation and this always, according to his equations, moved at a fixed speed. This meant, or so physicists then inferred, that there was a rest frame from which this speed could be measured. This, they thought, was the rest frame of the lumineferous aether, the medium through which light travelled and which filled all of space.
However, experiments failed to detect this rest frame. Einstein had the brilliant idea of positing that there was no rest frame but instead that the speed of light was an absolute motion. This led directly to the invention of special relativity through the examination of what simultaneity meant in this context. Similar ideas were also examined by Poincare around the same time.
Poincare also pointed out the Minkowski geometry of Einstein's spacetime but which was more fully developed by Minkowski. It is not hyperbolic geometry. In the latter, the metric is Riemannian and the sectional curvature is strictly a negative constant whilst in the former, the metric is Lorentzian and flat - the curvature vanishes. It is the analogue of flat space in Lorentzian signature.
It's worth adding that Einstein more or less identified the aether with spacetime. In fact, in the covariant description of Maxwells equations we use the tangent structure of spacetime to describe the electromagnetic field. So contra the textbook line of the aether vanishing into, well, the ether, it had been merely misidentified.
