Why are some areas still covered in fog after I saw the credits?

Because the credits after the Endgame are only the end of the first stage. The fogged squares on the ground map belong to later stages: almost all of them open in the second stage, which starts after the first ending, and the last two in the lower layer only after the third ending. It is not a bug. (A separate bug where cleared fog came back after reloading was fixed on October 15.)

What is the red panel in Order of the Sinking Star?

The game has no object called a panel; its English text never uses the word. What players most likely mean are the red signs: a handful of signs on the first-stage map that stay locked until you finish the Endgame, with the puzzles tied to them painted bright red on the map. After the first ending each one unlocks a piece (a lily pad, the purple beam, a door for a red stone, a mirror that points at the sky) that together send you up to the second stage.

Do the red signs clear the fog?

Not directly. The red signs build the way up to Antipode High. The fogged ground areas are then reached from above: 塞德莎 drops down from the High layer, or takes a teleporter, into each of them in the second stage. So the red signs come first, and the fog follows.

Can I clear any fog before the first ending?

A Steam player reports that one fogged tile on the west side of the Vault of the Beamcasters can be opened in the first stage with the purple beam; we couldn't confirm it in a video. Everything else waits for the second stage, so if you only need the four Gold Room teleporters for the Endgame, don't worry about the fog yet.

Does clearing the fog count for the Completionist achievement?

Completionist is for finishing every puzzle in the overworld, and the fogged areas hold overworld puzzles of their own, so expect to need them. 塞德莎 notes, for example, that the Heroes, Duplicated rooms only add stars to the sign on the map: nothing on the main route needs them, but they count towards finishing everything.

Order of the Sinking Star: How to Unlock the Fogged Areas and What the Red Signs Do (Spoilers Tiered)

Spoilers are folded. The first two sections are safe if you have finished the first ending (the Endgame in the Antipode). Everything after that names second- and third-stage areas; open a fold only for the area you are stuck on.

Short answer: the fog on the first-stage map is not a puzzle you can solve right away. Almost every fogged area opens in the second stage, after the first ending, and you reach it by dropping down from Antipode High. The red signs (what many players call the red panels) are the first-stage map’s meta puzzle: they stay locked until the Endgame, and solving them is how you climb to that second stage.

Before the first ending: leave the fog alone

On a first run the map has fogged squares you can’t reach. Steam players list the same ones: the far east of Heroes and Water, North; a strip west of the red area in the Vault of the Beamcasters; the east side of the Mirror Isles behind a rock; and part of Stones Illuminated (Steam thread).

You don’t need them for the first ending. The Endgame only asks for four active Gold Room teleporters (see the Antipode).
One possible exception, unconfirmed: a player reports that the westernmost fogged tile by the Vault of the Beamcasters can be reached in the first stage, and hints that you can’t get there from the square next to it and should use a purple beam. We have not seen this in a video.
Fog that comes back after a reload was a bug, fixed on October 15 (known bugs).

The red signs (“red panels”)

The game’s English text has no object called a panel. The red things players run into on the first-stage map are red signs: one by the Crossroads to the south, one in the red-floored area west of it, one in Heroes 3: Bard & Druid and one at the far east of the Mirror Isles, next to a cage. 塞德莎 walks past the Mirror Isles one early on and can’t use it (P6 21:03).

They unlock after the Endgame. They stay locked until you finish the first ending; after that, touching one opens it (P26 0:30).
The map paints them red. Every puzzle connected to a red sign is drawn in the same bright red on the overworld map, and small red arrows show where to push things (P26 0:20, 4:02).
What they are for: together they send the purple beam straight up from the Crossroads. Ride it and you reach Crossroads High, the start of the second stage, with the hidden Ascend achievement. The full solution is on how to ascend.

What each red sign gives you (light spoilers)

South: a lily pad you knock into the current. West: the purple beam. North: a door that lets a red stone out, and three red stones open the blocks at the north edge of the Crossroads. East (Mirror Isles): a mirror that reflects straight up. The steps are on how to ascend.

How each fogged area opens (second-stage spoilers)

In the second stage 塞德莎 goes back for the places he couldn’t reach, in a run of parts he marks 补 (“catch-up”). Most of them he enters by falling from Antipode High onto the ground map, or by a teleporter he had already switched on. Open the fold for the area you need.

East of the Mirror Isles, behind the rock

This is Mirrorable Mirrors, on the tiles 塞德莎 labels E3 and E4. He calls it a place “we couldn’t get to before”, right of the Mirror Isles, and drops in from the layer above in the second stage (P34 0:11). Solutions: Mirrorable Mirrors.

North-east, by Heroes and Water

These are the rooms of Heroes, Duplicated (N4E3, N4E4) and five extra Bard rooms of Heroes and Water. 塞德莎 reaches them by falling from the upper layer onto the first-stage map, using a teleporter he had already activated (P30 0:00). Solutions: Heroes, Duplicated and Heroes and Water, puzzles 30–34.

The Steam thread places one fog at the east of Heroes and Water, North. That this is the same spot is our reading of the map position, not something a video states.

West of the red area in the Vault of the Beamcasters

This is Alter Beam (W4), in the middle level of the West, left of the red puzzle area. In the second stage 塞德莎 drops to it from the West of Antipode High, back to the middle level; the new blue beam is introduced here (P38 0:20; title card at 1:24). Solutions: Alter Beam.

Under Stones Illuminated

This is the place 塞德莎 calls the Dragon Ruins, the area Squeeze, Dragon (S4W2, S4W3). His part 40 bundles “the places the first stage couldn’t reach” (P40 0:18). He says the fog under Stones Illuminated opens only after you beat the big level west of the dragon; then go up, take the teleporter and come down into it (P40 16:25, 16:50). Solutions: Squeeze, Dragon. Three more Stones Illuminated rooms come from a drop out of Double Beam Falling in Antipode High: Stones Illuminated, puzzles 24–26.

After this 塞德莎 counts three fogged squares left on the main map (P40 39:06).

The last ground fog: below Reflected Beams

This is Beams, Mirrorably (S4W5). In the part he titles “ground layer fully cleared”, 塞德莎 notes one fog left on the ground and walks to a cage that is now open; he isn’t sure which room unlocked it. He adds that to finish every room inside you need the rooms of Double Beam Falling in Antipode High first (P42 23:03, 23:29). Solutions: Beams, Mirrorably.

The last two fogs (third-ending spoilers)

The two fogs in the lower layer

The lower layer has two fogs that can’t be opened for most of the game. After the third ending a hidden path below Three at Once, reached from Crossroads Low, opens them; a recording waits behind them. The route is on developer levels and the hidden recording (P69 57:51).

What we could not confirm

What “red panel” means to everyone who searches it. The red signs are the best match, but late in the game there is also a red door and a red beam on the way to the Final Endgame (how to reach Endgame Low).
The first-stage purple-beam tile by the Vault. Only reported on Steam.
Which room opens the cage to Beams, Mirrorably. 塞德莎 says he doesn’t know either.
Whether every fogged square the Steam players list is exactly the area named above. We matched them by map position.

See also: achievements (Ascend, Completionist) and how to reach Endgame Low.