I've had Houdini analyze some of my recent games.

While it tells per each move made which side it thinks is doing better, e.g. +0.38 if White is doing a little better, it doesn't tell me what it thinks is the strongest move for the side to move.

Nor does it tell me *why* White is slightly better.

For both of these functions - what is the best next move and why one side is faring better than the other isn't that what one would need a human chess coach for? Just pondering. Can one really learn much of value from machine analysis?

The computer should give a short variation if it finds a move that's better than the one played. How often it does this depends on what your evaluation threshold is.  eg. if your threshold is set to 0.3, the computer will only suggest a better move if it's evaluation is better than the original move by 0.3

To understand why White is better you have to use your own skill in analyzing a position, and if that's not up to the task, the help of a coach or stronger player. A computer will tell you what it thinks is objectively the best move, but that doesn't guarantee it is the best move for a human player...computers have no fear and will enter razor-sharp complications or difficult positions without second thought if it evaluates the resulting position as even minutely better

So, for example, in this game, after 14.Kxg2, Houdini says Black has an advantage of -5.28.

I have no idea how it reaches that conclusion, as the sides look roughly evenly matched. Black is a pawn up, that's all.

Not sure why Houdini isn't giving you a continuation, but my engine simply points out that after 14. ...neg4 White can't prevent the fork on e3, winning at least another pawn and the exchange.  That's four points of material advantage, plus White's king is still rather uncomfortable.

You must be doing something wrong. What GUI are you using, and which settings?

When you let engines analyze a position, they should give you the best continuation ('Principal Variation') for an ever increasing search depth. The evaluation they report will be the static evaluation (roughly material count + king safety + pawn structure) of the position at the end of this PV. You can even set most engines to 'MultiPV' mode, where they don't only give the score and continuation after the best move, but after a number of moves.

If your score is -5 at depth 12, while in the position itself you are only a Pawn behind, it means that the PV ends (after 6 moves for each side) in a position where you are a Rook behind, when both sides play their best moves. In other words, even against best play black can force the gain of a Rook (or equivalent material), and the PV shows you how this is accomplished.

In WinBoard, for instance, when you load the game, step to the position of interest, and set the engine to Analysis Mode, you will see all the PVs (starting with the best move) in the Engine Output window. When you right-click a PV there, and keep the mouse button down, you can walk through the PV by moving the mouse 'vertically', seeing on the board how the engine think the position will evolve on best play, until you reach the end of it. That position should be the one that was responsible for the score. (And if you release the mouse button there, the engine will continue to analyze from there. If you don't want that, step back to the beginning of the PV before you release.)

Puters are no good nincompoops!

I've had the chess.com computer analysis done a few times, and I find it pretty enlightening.

And when I *really* want to give a game thorough study, I pick over it with Fritz. But that's a lot more hands-on, involving me going back and working through parts of the game against the engine & poring over the mountain of information it spews out.

See what moves the computer suggests!

chessmaster says this

Leads to 14...Neg4 15.Kg1 Nxe3 16.Qf3 Nxf1 17.Rxf1 Be5 18.Bxe5 Rxe5 19.d4 Rg5 20.Nd3, which wins a bishop for a pawn. Better is Nxg2.

mentor lines says -4.2

I let chessmaster think for 60 seconds as black
The Chessmaster recommends: Knight at e5 to g4.

Analysis: You move your knight at e5 to g4, which attacks White's pawn at e3 with two pieces and blocks White's pawn at g3. White counters by moving the king to g1. Your knight captures pawn, which forks White's queen and White's rook at f1. White replies by moving queen to f3, which moves it to safety. Your knight captures rook, which threatens White's knight at d2. White responds with rook captures knight, which removes the threat on White's knight at d2. You move your bishop to e5, which attacks White's bishop. White answers with bishop takes bishop. Your rook captures bishop. White replies by moving pawn to c4. You move your pawn to c6. White responds by moving pawn to d4, which attacks your rook at e5. You move your rook at e5 to e8, which moves it to safety. White counters with the knight to e2.

As a result of this sequence of moves, you win a rook, a bishop, and a pawn for a bishop and a knight. Additionally, White's attack potential is decreased. Also, your pawn development is a little stronger, and White's pawn structure is a little weaker

Chess King Pro. If I get it to analyze the entire game I get no continuations from the engine, only assessments of positional strength/weakness.

Never heard of Chess King Pro... Use something better, then. When you analyze an entire game with WinBoard, (Mode -> Analyze File, or in more recent versions Mode -> Analyze Game) it can save the entire PV as a comment with every position.

But it is usually more enlightening to analyze specific critical positions, so you can use a much larger depth without wasting too much time on the rest of the game (e.g. positons where you could recapture in only one way...).