Why is space and time spoken of as one thing

There are different words for different aspects of space. For example, consider: length, width, and height. Other words include depth and breadth. We can speak of them as different things if we choose to, but we generally consider them to be part of unified concept of space. Why?

It's because we understand that these words just pick out measurements along specific directions in space set by context or the speaker's viewpoint, and that there is no intrinsic difference between those directions. Depending on which way they are facing, what's length to one person is width to another.

More formally, there is a symmetry between directions in space. We can rotate freely and the intrinsic properties of space look just the same, e.g. the Euclidean metric (infinitesimal distance formula)
$$mathrm{d}s^2 = mathrm{d}x^2 + mathrm{d}y^2 + mathrm{d}z^2$$
is invariant under the orthogonal group of rotations, $mathrm{O}(3)$. Together with spatial translations (which are like rotations about a point at infinity), this forms the Euclidean group of isometries, $mathrm{ISO}(3)$. Because of this symmetry that "intermixes" directions of space, we think of those different directions as being unified one into one thing: space.

In special relativity, spacetime has a Minkowski metric (in units of $c=1$)
$$mathrm{d}s^2 = -mathrm{d}t^2 + mathrm{d}x^2 + mathrm{d}y^2 + mathrm{d}z^2text{,}$$
and it is invariant under the Lorentz group $mathrm{O}(1,3)$ that act like rotations in spacetime; together with the translations (in space and in time), the full isometry group is the Poincaré group. In fact, the usual Lorentz transformation (say, boosting along the $x$-axis) is exactly a rotation with by a hyperbolic angle in the $tx$ plane.

That's why people began to think of spacetime as one unified whole. The reasons are essentially the same as why people think of space as one unified whole despite it having different directions: there is a symmetry that "intermixes" those directions, albeit slightly differently in spacetime because of the different sign of the temporal and spatial directions in the metric.

In general relativity, this is generalized further, but the details aren't immediately relevant here.