The Ethereal Plane is another dimension, described in more detail on Pages 48 and 49 of the DMG. It may help you to replace the term "plane" with "dimension." The ethereal plane borders several other planes, and is often described as a muted, foggy outline of whatever exists on the plane that it borders.
So, take for example that you are in a corridor inside of Castle Ravenloft. There is a spot in the ethereal plane which looks exactly the same as the castle corridor, except it's all fuzzy looking and grey and foggy. Any creature on the ethereal plane is invisible to the stuff in the castle corridor, but the creatures in the corridor are visible to the ethereal denizens. Creatures on the ethereal plane also feel no gravity, and can move through walls easily.
The ethereal plane is used very often then for travel, as well as espionage, since travelers can move through walls, as well as spy on people on the material plane without worry of being discovered. Ghosts and other such incorporial undead can move into this plane at will (which is what allows them to turn invisible).
If a character can use true sight, they can see into this plane. So if anything is spying on them using this plane, they will see it. They can also see ghosts and other such spirits that try to hide from them by moving into this plane. They will also see things that were hidden in the ethereal plane for safe keeping.
As for the effect of this on your campign, that really depends on how much you use alternate dimensions as a DM. If you don't use the ethereal plane for very much, the spell just means that the PC can follow ghosts and other such things, as you have noted. If you like to put spies in there, or other such hidden things "stashed away in the fourth dimension", then the PCs will notice those things. The effect is pretty minimal, since your players would acctually have to shift themselves into the ethereal plane in order to affect anything you put on that plane.
As noted by Andrewk, Ravenloft is filled to the brim with trapped ghosts. Even if they don't directly interact with the players, it can be a powerful narrative/aesthetic device for the PCs to see so many souls in the ethereal plane as they move through the castle.
Best Answer
See into the Ethereal Plane yourself
The Ethereal Sight mode of Divination Wizards' 10th level feature The Third Eye grants the ability to see into the Ethereal Plane as an option.
Two spells explicitly lets a character see into the Ethereal Plane:
See invisibility (2nd level; Range: Self)
True seeing (6th level; Range: Touch)
Then of course you can gain truesight, which per the description in the Monster Manual (page 9; or PHB p. 185):
Three magic items grant truesight as part of their effect (I'm not going to list magic items from modules as that would make the list essentially unbounded):
A Gem of Seeing (rare, requires attunement) will grant truesight for 10 minutes for each of its 3 charges.
A Crystal Ball of True Seeing (legendary, requires attunement; a crystal ball variant) adds truesight to the scrying spell cast through it.
The Eye of Veccna (artefact, requires attunement) grants truesight (no range listed) as one of its many benefits. No downsides, promise.
If applicable to your game, the Third Eye Nomad Dicipline from the UA Mystic Classgrants truesight. Alternately, there are monsters with it and spells that let PCs take the form of a monster. As there are no beasts with truesight (bummer, right?) we need to be able to take any form. True polymorph and shapechange are good bets and you can find a list of monsters with truesight here.
Travel into the Ethereal Plane
If you are on the Ethereal Plane, you can see other creatures on it (and interact with them generally).
Etherealness (7th level; Range: Self). This spell transports you to the Ethereal Plane and if cast at a higher level will let you bring some willing creatures along. (Horizon Walker Rangers get the ability to cast a limited version of this spell at 7th level.)
Blink (3rd level; Range: Self). This spell will transport you to the Ethereal Plane, but with the side effect of transporting you back at the start of your turn. It's a bit weird, but it works.
"Companions" to see into the Ethereal plane
If you don't need to see into the Ethereal yourself, but can be sated by a companion (i.e. another creature under your command/control) doing it, there are a few ways for that too.
Mordenkainen's faithful hound (4th level). The phantom watchdog created by this spell can see into the Ethereal plane and alert you of any activity there.
Conjure celestial (7th level). When cast at 7th level, this spell can summon a Couatl (CR 4), which has truesight; when cast at 9th, it can summon a Battleforce Angel or a Felidar, both with truesight and from the Guildmaster's Guide to Ravnica (ask your DM about availability).
Summon greater demon (4th). When cast at a higher level, this spell will let you summon a demon of CR one higher that spell level. Notably, Glabrezu and Master of Cruelties (again from Ravnica; ask your DM) are CR 9 demons with truesight and so can be summoned with a 8th level casting.
Planar ally (6th level). This spell can summon a celestial, elemental, or fiend to help you. You will have to bargain for the service, and again see this list to find options.