There are a lot of ways to analyze a piece. Analysis is just a way of doing detective work to understand what the composer has done, and we can use a lot of tools. We can look at the changes from one key area to another, the patterns of individual chords, the melodies or "motivic" patterns, the rhythms and meters, and other things. For a metaphor, if we were trying to analyze the architecture of a building, we might think about the engineering of its structure, the shape of its silhouette, the ornamental details, etc.
It helps to learn about chords and keys so you can do "harmonic analysis," since this is analogous to the engineering of the architectural analogy: the key areas and chords are the structure that "holds it together." It also helps to learn how to read music so you can study the score; this is the "blueprint"! But even without these areas of knowledge, there are a lot of simple questions you can start with (and everyone would be smart not to skip). Where does the music make big changes? In this particular movement, the orchestra, especially the basses, join in and play loudly in only a few moments, and others are marked by only upper strings, playing softly; these help us find the major sections. You can even recognize some melodic "shapes" just by looking at the score and noticing where the melody moves from note to note in small steps or large leaps, where it holds notes for a long time or has many active, small notes.
For instance, look at the soloist's opening material:
I've drawn dotted lines between some notes. Notice how the first two big shapes, in green, follow straight lines through several quick notes, moving step by step down the scale. They're separated, though, by a leap that jumps up five steps at once, marked in red. Meanwhile, in the next measure, instead of two main shapes, we get a zig-zag, "sawtooth" appearance, as the melody jumps up, then steps down, and repeats that small shape five times.
It's also helpful that you've referenced a video that visualizes the audio with similar lines and shapes.
So what do we make of this movement? Harmonic analysis can help us explain the overall story. We start in C major and we end in it, but in between it visits some other "keys" (or more accurately, "tonal centers and modes," since it also plays around with major and minor). Taking a broad view of those changes, we see:
- We start out in C major (here's the start in the video, or to follow along in this copy of the score, we start on p. 9)
- At the end of the soloist's second section, top of p. 10 in the pdf, we're in G.
- By the end of that section though (p. 10 second system) we're in D minor.
- The orchestra embraces that D minor and the soloist starts out the next section in it, but they finish that section by cadencing in A minor.
- The soloist's next brief section—well, honestly, I haven't done the hard work to figure out what it's doing; it gets very tonally adventurous in a very small space. But it all ends happily ever after, because by the end of that section (p. 11 system 3) we get a nice big, fat, juicy cadence back in the C major key that we started in. This is the happy ending to the big story of "There and Back Again": We left home, we had some adventures, but we ended up back where we started. The piece isn't over yet, but we don't want to get too far from C major from here until the end, now that we've gotten back to it.
- So the soloist starts the next section over an F chord, but it doesn't mean we've left the key of C; F is the "IV chord" in C, a useful way to stretch the harmony out and kill time without really going anywhere. Sure enough, we get another cadence on C a few bars later (top of p. 12), and another at the very end of the movement.
So to zoom way out and summarize that tonal structure, we have "C - G - D(m) - Am - (something) - C." The beginning of this pattern suggests the Circle of Fifths, at least because the tonal centers are moving up a fifth each time (even if sometimes they slip into minor).
Okay, now let's look at melodic material. We already have an image above of the first notes that the soloist plays. Let's call this "Pattern 1." It's characterized by a sustained note from which it falls in downward stepwise motion, and by those "zig-zag" shapes in orange which move downward each time. The whole section is very short, only two measures; it seems like just a sort of "introduction" before anything serious, but let's keep that shape in mind.
The next solo section looks like this; let's call it Pattern 2:
It's both less "zig-zaggy" and less stepwise than Pattern 1. It undulates gently up and down, mostly moving in intervals that are small but are more than a neighboring step. Instead of tracing lines, I've circled a few important notes: E, A, D, G, C. Compare this to the progression of tonal centers that the entire movement outlines—it's the same thing in reverse! (Especially if I can find an E somewhere in the mystery section.) In particular, the melody arranges these pitches so that it goes from E up to A, down to D, up to G—and, well, then up to the next C as well, but for a while there we had a big zig-zag going.
Let's consider the soloist's next big section (in the full score, this is the top of page 10):
At first glance, this is new material. But there are similarities to Pattern 1: We start with a sustained note—sustained much longer this time—and fall away from it in stepwise motion. And, in the orange box, we see the same zig-zag pattern; this measure is clearly the last measure of Pattern 1, but mutated to fit a new key. Okay, so let's say this counts as Pattern 1. In fact (spoiler alert), we'll see more things that look just like this version, so maybe this is the "real" Pattern 1, and the first appearance was just an abbreviated version.
The next section starts clearly with Pattern 2:
For the start of the next section, don't let the first few notes fool you. It's Pattern 1; it just throws in a few notes to "lead into" it, but in the second measure, there's that sustained note and downwards scale.
At the bottom of that screenshot we see that the next section is another carbon-copy occurrence of Pattern 2.
That leaves only the mysterious final scrap of melody that the soloist plays over the orchestra's closing four bars:
At first I was ready to dismiss this as just a bit of quasi-improvised ornamentation, a final flourish. But look at the circled notes: A - D - G - C - F (one could arguably even extend the pattern with the next-to-last note, B). This is Pattern 2! We're moving up a fourth, down a fifth, up a fourth, down a fifth, etc. It's been "disguised" by filling in all the gaps with ornamental notes. One could actually play the undisguised Pattern 2 (transposed to C major) over these notes and it would fit nicely. You were right—you have been "hearing this melody all along"; it's just that it was in a different rhythm. In fact, you can even find it hiding in the harmonic structure of the overall piece, magnified into huge proportions, to "slow" to "hear," hiding by being too large to notice.
That's my start at just an initial assessment of the melodic and harmonic building blocks. More time or more expertise could probably reveal more (and probably has before now!).