Why Syzygy made this queen sacrifice?

[kinderchocolate]

FEN: 8/8/4K1p1/8/7k/3R4/8/5rq1 b - - 0 1

[d]8/8/4K1p1/8/7k/3R4/8/5rq1 b - - 0 1

Using the complete 5-men Syzygy collection:

./stockfish-8-64
setoption name SyzygyPath value <TB Path>
ucinewgame
position fen 8/8/4K1p1/8/7k/3R4/8/5rq1 b - - 0 1
go infinite

.......
info depth 4 seldepth 6 multipv 1 score cp 12850 nodes 726 nps 181500 tbhits 18 time 4 pv g1e3 e6d5 e3d3
.......

Why does Black make ...Qe3+? Why not ...g5? Does that make sense?

When I checked (https://syzygy-tables.info/?fen=8/8/4K1 ... _b_-_-_0_1), it's not clear to me why ...Qe3 was played.

[Guenther]

FEN: 8/8/4K1p1/8/7k/3R4/8/5rq1 b - - 0 1

[d]8/8/4K1p1/8/7k/3R4/8/5rq1 b - - 0 1

Using the complete 5-men Syzygy collection:

./stockfish-8-64
setoption name SyzygyPath value <TB Path>
ucinewgame
position fen 8/8/4K1p1/8/7k/3R4/8/5rq1 b - - 0 1
go infinite

.......
info depth 4 seldepth 6 multipv 1 score cp 12850 nodes 726 nps 181500 tbhits 18 time 4 pv g1e3 e6d5 e3d3
.......

Why does Black make ...Qe3+? Why not ...g5? Does that make sense?

When I checked (https://syzygy-tables.info/?fen=8/8/4K1 ... _b_-_-_0_1), it's not clear to me why ...Qe3 was played.

The answer is already in your post. You used only the 5men syzygy.
With Qe3+ it can force a (known) 5men win one move later, by either
deliberately losing its Q, or winning the opponents rook.

I don't know the exact implementation in SF, but probably the search
would need to find a score above the given 128.50 for not playing this
winning sac.

Humans also do this regularily to save time, when the opponent
refuses to resign. (mostly to reach dead won pawn endgames - though
usually not in positions with that huge material up and no complications)

[syzygy]

I don't know the exact implementation in SF, but probably the search
would need to find a score above the given 128.50 for not playing this
winning sac.

And the engine would have to find a way to get a >128.50 score without converting into a 5-piece position. Because once something is captured (including the black rook), SF will score the position as 128.50 minus a small number proportional to the number of plies it takes to reach the position.

So effectively black can defend against a >128.50 position by endlessly checking the white king with his rook. If white takes the rook, the position is 128.40 or so.