How does the slow spell actually work in D&D 5e?

The description states: "You alter time around up to six creatures of your choice in a 40-foot cube within range."

Is this talk about their perception of time, so they see everything moving really fast? Or does it affect the entire AOE space?

If you're altering time I would assume objects could be affected, but the spell mentions only creatures - but that seems odd for the ability to alter time in a 40-foot space.

Spells Only Do what they Say They Do

The Slow spell only dictates that it affects a creature's movement speed, their ability to use actions and reactions, the number of attacks they can make in a turn, and a creature's spellcasting abilities. The spell doesn't specify effects on projectiles or other non-creature objects, so Rules as Written, the Slow spell does not affect any of these things.

Be wary of giving extra effects as a Rule of Fun option

Being able to slow down projectiles would probably dramatically improve the effectiveness of a spell like this, since it would potentially give defenders the ability to avoid ranged attacks entirely, depending on how you chose to rule the effect of such a phenomenon. This would make the spell more powerful than intended, and Slow is already considered to be a quite powerful spell. Other things like slowing down falling rocks could have very significant consequences for the game.

How to interpret the line "You alter time around up to six creatures of your choice"

In principle, there's no difference in 5th Edition D&D between flavor text and mechanics.

In principle.

In practice however, there aren't a lot of mechanics that specifically care about a creature's personal "flow of time", so to speak, and even for features that do, the Slow spell doesn't specify how exactly their personal sense of time is affected, except to specify the specific mechanical constraints laid out by the description.

So it's probably not worthwhile to pick apart the possible mechanical semantics/implications of that line. Stick to what the spell actually says it does, and maybe encourage some roleplaying consequences, like forcing your players/characters to speak in a deliberately slowed down manner, or being surprised by the alacrity of a character moving at normal speed adjacent to them.

Mechanics

You alter time around up to six creatures of your choice in a 40-foot cube within range. Each target must succeed on a Wisdom saving throw or be affected by this spell for the duration...

... A creature affected by this spell makes another Wisdom saving throw at the end of its turn. On a successful save, the effect ends for it.

So first, assuming you have the necessary components and are otherwise able to cast, you pick a 40 foot cube entirely within your 120 foot range. After that, you select up to six creatures within that cube. Then, each that fails its save is subjected to the mechanical penalties described in the spell.

As Xirema's answer points out, D&D spells only do what they say they do. In this case, the spell says that it affects up to six creatures, only applies its mechanical penalties to those creatures, and only affects those creatures that were selected and failed their Wisdom save. It does not affect all time within the entire 40-foot cube: the cube is just a limitation on how far apart the selected creatures can be from each other when the spell is cast. It does not affect the entire 40-foot cube. And by RAW reading, none of the mechanical affects are applied to the space around those creatures either - that appears to just be flavor text.

The rest of the spell's description is the specific mechanical affects on the creatures that failed their saving throws.

What About the Fluff?

At this point, I'm moving out of objective analysis and into subjective interpretation.

An affected target's speed is halved, it takes a -2 penalty to AC and Dexterity saving throws, and it can't use reactions.

It might seem strange to quote a detailed mechanical effect to interpret the fluff here, but that's what I'm doing! Notice the -2 penalty to AC and Dex saving throws. These only make sense to me if the space around the target is unaffected, and it's just the target itself that is Slowed. Judging by this line, I would rule that the target itself moves through time more slowly than its surroundings do. If anything entering its space were similarly slowed, then the target might not suffer that penalty, but that's not what we see.

Because of this, I would personally rule that, to the affected creature, all the rest of the world appears to be moving more quickly, even objects or other creatures entering its immediate proximity. Since the spell doesn't say anything about an altered falling speed or slowing of nearby objects, I would not apply such things.

Now having said that, one of the central tenets of D&D 5th Edition is to do what's fun, and the rules really are only there to give a structure to guide the DM. Those rules do not have the final say.

So, you can interpret the spell as I have, or you may end up with a different interpretation altogether! But however you interpret it, make it consistent, make it fair, and make it fun!