As usual, the main ideas here are wildly speculative, written down as food for thought. I am not claiming to be a cosmetologist or astrologer.
For the purposes of this post, let's suppose that the universe is ultimately discrete. By this, I mean that when you get down to its most fundamental level, there are "pixels" in one way or another, some ultimate underlying property of everything which is atomic and indivisible, digital and not analog. If the uncertainty inherent in quantum mechanics drives the deepest level, then the rest of this may not apply, but it is certainly possible that there are deeper underlying forces yet to be identified, so we don't know yet.
Note that the discreteness of the universe does not preclude space or time from being infinite (although it is arguably an infinity of a lower order). However, I'm about to suggest here that it does link the finite-ness of space and time inextricably: either they are both infinite, or both finite.
Consider a discrete universe with a finite volume, but lasting forever. Any such universe could be reasonably treated as an enormous finite-state machine. No matter how vast it might be, there would be only so many possible configurations it could take on before repeating itself.
If its physics are deterministic—configured such that any given arrangement of "pixels" (or "atoms" or "bits") necessarily defines its successor—it would use only a tiny fraction of all possible states. Even if there is some purely random influence at that level, and it could conceivably reach every single possible state, there would still be a limit to the number of possible states.
Granted, it would be enormous; for example, assuming a universe comprising %10^100% bits, there would be %2^(10^100)% possible configurations for it to be in at any given moment. Note that this is a big fucking number; we're talking %~30000...0% possible states, where the %...% is replaced by about %1,000,000,000,%%000,000,000,%%000,000,000,%%000,000,000,%%000,000,000,%%000,000,000,% %000,000,000,% %000,000,000,% %000,000,000,%%000,000,000,%%000,000,000% digits.
If we consider its progression over time, we're looking at all the permutations of those configurations, representing an upper bound on every possible narrative the universe could follow, of which there would be %2^(10^100) !% configurations. The number above, which we could not write because it has more digits than the number of electrons in the observable universe, is the number of digits of this new, incomprehensibly larger number. (Yet still finite, so overall, pretty small as numbers go.)
But I digress. So let's focus on the deterministic case, which seems likely to be the correct one in a discrete universe. If we have a finite number of bits, that means that sooner or later, we're bound to come back to an exact state that already occurred in the past. That point necessarily defines a set of states through which the universe will cycle, over and over, forever.
Each cycle will only take a finite amount of time. This means that time cannot truly be termed infinite, since there would be absolutely no reference point by which a state occurring in one cycle could be distinguished from the same state in the next cycle. Time would be a loop of finite length. Thus: in a deterministic universe, finite space implies finite time.
What was less clear to me is whether the converse follows, that finite time implies finite space. The symmetry of space and time biases me strongly towards thinking this is probably the case, but let's look at it.
In the finite-space situation above, I claim that time is effectively finite because there are a limited number of distinct states, rendering it meaningless to speak of any time beyond that necessary to differentiate each of them. In a finite-time universe containing infinite space, we might be tempted to look for the same general pattern; a finite cycle such that regardless of distance traveled (rather than time elapsed), there are a limited number of possibilities one can end up with.
Let's consider our infinite-space universe as an infinite number of bits spiraling out from a point-source observer (you). Pretend there is no speed of light limiting things in this universe. Even with a finite lifespan, the universe would still very much be spatially infinite, both in the sense of being able to observe every single bit in that infinite string, and secondarily the possibility of traveling an arbitrary distance within it so that you are now surrounded by an arbitrarily large amount of completely different information than wherever you started. This is clearly not analogous to the finite-space situation above.
How might we fix that asymmetry between time and space if we were designing a universe and wanted to keep things simple and balanced? Mix in relativity. With the speed of light governing things, any observer is now limited to a light cone demarcating a strict upper bound on the observable universe; this is equivalent to establishing a finite limit to the number of bits and consequently number of possible states perceivable for a finite interval of time.
In that sense, the invariant nature of the speed of light seems almost specifically tailored for the purpose of linking space and time together. All of the fun relativistic side effects you end up with are logical necessities conspiring to limit the information available starting at any point in space over a given period of time. Thus: in a deterministic universe, finite time implies finite space—so long as you have relativity!
Finally, the logical corollary to these assertions, taken together, is that infinite space implies infinite time and vice versa.
Conclusion: Maybe that's why relativity.