Right, ZauberTigga, here is my attempt at explaining the rules:

nice comparison vanatteveldt, but unfortunately the formula "movement cost -> visibility range" doesn't always add up exactly. Generally the mini-Scouts walk backwards and can choose between paths of equal shortness.
The plain tile 1 NW of Pasargadae (E1) is visible because the path of the mini-Scout goes E1-E2-E3-D4. They always just walk 1 tile at a time and then decide to step on the tile which is closest to the current beeline. So standing on E1 they choose between 3 tiles (D1, D2, E2) and E2 wins because it is closest to the current beeline (orange arrow). Standing on E2 they choose between D2, D3, E3 and now both D3 and E3 are possible paths, since they have the same distance to the current beeline (green arrow). But D3 (forested hill) blocks the vision, so E3 wins.
From the tile 2W of Susa (A6) the path can only go through B5 (closest to the red arrow beeline) which is forested and thus the vision is blocked.
The determination of blocked/unblocked vision works by comparing "seefromlevels" and "seethroughlevels" of tiles. seefromlevel (SFL) is 0|1|2|3 for water|plain|hill|peak. seethroughlevel (STL) is equal to SFL but increased by 1 for tiles with forest/jungle.
The Sentry Keshik is standing on a hill so it's SFL is 2 (green number in D4 --> "SFL0"). The tiles in the innermost ring are always visible, but can block vision for the tiles in the 2nd and the outer ring. The white / light blue numbers are the STLs for the inner / 2nd ring (--> "STL1 / STL2"). The yellow numbers are the SFLs for the outer ring (--> "SFL3").
The outer ring has special rules. For units without the Sentry promotion, the 2nd ring naturally becomes the outer ring.
Tiles in the 2nd ring will be blocked by tiles in the inner ring with a STL1 > SFL0 (the forested hills in C3 and D3 block the tiles north of them). Tiles in the outer ring are visible if either SFL0>STL2 (E1 is visible because SFL0 = 2 > STL(E2) = 1) or their SFL3 > max(SFL0, STL1, STL2) (the peak in A7 would be visible even if B6 had a forest).
The tech Optics increases the SFL of water tiles by 1 --> you can build taller ships

.