To preface this, I am very new to chess. I knew the rules for years but I started actually taking the game more seriously and learning strategies just over 2 weeks ago.
One thing I consistently see in high elo games is that they trade same value pieces constantly. What am I missing? From my point of view, they are getting no value and just wasting a turn...
There are a few classic reasons which tend to make exchanging pieces favourable. None of these are absolute; in any given position, short-term tactical considerations may dominate. But you might like to keep the following themes in mind which can favour exchanges:
(1) You have a material advantage.
(2) Your opponent has a space advantage (your position is cramped).
(3) Your opponent has dangerous prospects of an attack on your king.
(4) The structure of a position (most typically, the pawn structure) favours you in the endgame.
In addition there are many familiar patterns whereby an exchange creates an advantage for you. For example, the opponent has to accept doubled pawns by a pawn recapture; or the opponent has to give up the right to castle by recapturing with the king (this often applies to queen exchanges in the opening). Or a particular exchange favours you, because your piece is passive / badly placed, while theirs is active / well placed.
(And just to complete the answer: as several people pointed out already, exchanging doesn't typically "waste a turn" because the opponent also uses a move to recapture. If for some reason forcing an exchange does involve giving away the initiative, then indeed that will often be a factor against it.)
Trading pieces thoughtlessly is bad. As you say, it wastes a move. Good players trade pieces when it gives them an advantage like exchanging a passive piece for an active piece or when a piece is threatening bad things or when the opponent has spent several moves developing the piece.
Pieces can also be exchanged to change the nature of the position, to get a position the player is more comfortable with or that the opponent is less comfortable with. An example could be exchanging queens against a player who likes to attack.
If a good player exchanges pieces and you can't see why that doesn't mean it was a poor exchange. It might just be that you aren't good enough to understand why.
Not all advantages in chess are material. You may want to trade pieces to gain other sorts of advangages. You're often not really wasting a move, since your oppoenent will also waste one of their own moves to recapture.
Also, the "value of pieces" is an abstraction that may not correspond to what pieces are actually worth in a specific position. For example, after 1.e4 e6 2.d4 d5 3.e5, Black would probably be happy to trade his light-squared bishop for any of White's minor pieces, but he would definitely not trade the f8 bishop for a knight!
In summary, as other answers have already pointed out, trading pieces for the sake of trading pieces is not a good idea. But trading pieces to get some other kind of advantage is a different story
James Martin has explained possible strategic advantages for one side to trade a piece. But that raises another question: if trading a piece is favorable for one side, then it must be unfavorable for the other. So, why would they allow the trade?
In some cases, avoiding the trade may be bad tactically. Often, this is because a trade of a piece actually wins a tempo, not loses!
[FEN "r1bqkb1r/pppp1ppp/2nn4/1B2p3/3P4/5N2/PPP2PPP/RNBQ1RK1 w kq - 1 6"]
6. Bxc6 (6. Ba4 exd4) dxc6 (6... bxc6 dxe5) 7. dxe5 Nf5 8. Qxd8 Kxd8
The classical position from Berlin variation of Ruy Lopez. White bishop is attacked; if it moves, Black has time to deal with a threat to their e5 pawn, and ends up a pawn up. But instead the bishop captures the knight, in one move dealing with the threat to itself and forcing Black to answer. The tempo thus gained is well-spent capturing on e5.
On move 8, trading queens in itself benefits Black, as it diminishes White's attacking chances. But it deprives Black from castling, which is good for White. And avoiding it would cost White an important tempo. A combination of these three factors lead to the trade being the best move for white, while allowing the trade by 7... dxc6 is considered the best move for black.
In other cases, avoiding a trade is a strategic concession:
[FEN "4r2r/pp1k1ppp/2p5/3p4/3P4/2P5/PP3PPP/R3RK2 w Q - 0 1"]
There is only one open file for the rooks in the position, hence, removing one's rook from it will concede it to the opponent. Nether side really wants it, so the rooks will likely be traded.
Ultimately, if a player does something, it's because they think all of the alternatives are worse. There are a variety of ways that piece trading can occur, but one situation is where one player's (say, White) piece is presenting a threat, and so Black trades for it. Taking the piece costs Black a turn, but if White wants to recapture, then White has to spend a turn doing so. So Black didn't have a net loss of a turn. Furthermore, otherwise dealing with the threat would also cost at least a turn, and quite possibly material and/or position.
Typically you'll trade pieces when you have a material advantage and want to reach an ending where this advantage will be maximized, simplifying your win. The nonpareil World Champion Jose Capablanca was noted for gaining a pawn advantage and trading into a winning ending. If you don't have a material advantage but are in a cramped position, trading down will help salvage the game, lessening your opponent's attacking possibilities. This would be particularly true if his pieces were around your king position and yours were not in an ideal position to counter them. Conversely, if you are attacking in such a position, trading may be desireable to remove a defender. When my opponent has castled on the king side, I try to remove his king knight on f6 (or f3) to make his h7 (or h2) pawn more vulnerable.
A simple example from 15 minutes ago. I had his king-side knight pinned against his queen. He put his bishop in the way, then moved the knight to h7 to drive off my bishop, or to trade bishops. What I don't think he noticed is that after the bishop trade I took his knight with my other bishop--he has to recapture with his rook and thus can no longer castle king side.
