I've never had to explain them beyond simply stating - once - what they are (a chance to avoid or mitigate something bad, usually) and how they work (roll a d20 when I ask you to, higher is better, and let me-as-DM sort it from there).
The saving throw matrix is DM-side info, despite various attempts in 1e and-or 2e to put it player-side.
The treadmill is a product of the change in purpose. The treadmill allows characters to face greater threats with similar chances of success, while easier threats are bypassed more easily. In the old style, failing a save was usually catastrophic, so by having them increase slowly over time represented the character's experience. Using the old style with mitigating saves would mean that most players would stop caring about saving throws as they level.
This is just incorrect. There's a big difference between thog the half troll barbarian not caring about fort saves or merlin the wizard not caring about will saves at high level vrs not caring about any saves because Thog is going to get creamed by will saves & merlin is going to fail a fort save just by being near one. 2e was a little different in that it became easier to save (roll a d20 hiugher than number listed on that) but a failed save was almost always awful.
I'm in-between Middle and Modern. Old School does some things well, but its not my favorite.
Since others are posting their alternatives...
This assumes the six Ability Scores of D&D and the 5e static AC vs variable saves.
Deflection: Str or Dex + Armor (most attacks with melee or ranged weapons, the only static one, call it AC, Block, Parry, or whatever you want)
Fortitude: Str or Con + d20 (resist poisons, push and pull, crushing, falls)
Reflex: Dex or Int + d20 (avoid AoEs or particularly quick attacks like Lightning Bolt)
Will: Wis or Cha + d20 (mostly fear, charm, command, and things like that)
Mind: Int or Wis + d20 (thwart illusions and psychic attacks)
Death: Con or Cha + d20 (prevent dying when dropping to 0 HP and some nasty necrotic spells and abilities, at high levels you realistically only fail 0 HP death saves by taking damage)
All of these also add proficiency except Deflection unless you want to redo the armor table.
Saving throws are a bad concept that should have stayed in the place the worked - wargames.
They weren't a terrible fit for early D&D, but literally every edition except 4E they've become a worse fit for D&D. To the point now where they seem kind of idiotic, apart from Death saves. So I just want them to die at this point.
I like the "modern" approach, but I wish they had different names than the abilities. Having them be the same is one of those things that seemed beginner friendly to some WotC designer's gut sense of what makes things accessible, but actually just causes endless confusion at table and requires extra verbiage in manuals. I'm not sure I've ever been in a 5e session where someone didn't confuse ability check and saving throw.
I also think the lack of separate names for the saves derived from various abilities caused the conceptualization of what the respective saves really mean to be a bit weak.
But the basic "proficiency, based on starting class, in one common and one rare save, with advances based on proficiency bonus in 5e is a nice mechanic, it just needs less clunky terminology. I know some people advocate for just dropping the rare saves, but they really don't hurt anything and give characters whose primary stat is associated with a rare save occasional chances to shine.