Some musicians are content merely to play the notes on the music in front of them correctly, and are not curious to learn anything about the notes that other people in the ensemble also play, or how those notes affect each other. Other musicians want to know the "big picture" and understand how the composer has constructed the entire piece that the ensemble is playing. Still other musicians want to go beyond playing one instrument and to become a conductor, arranger, composer, or music teacher. Conductors, arrangers, composers and music teachers need to understand how to construct an entire composition, not just how to read the notes in one part correctly.
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As a trombone player, you are accustomed to being handed a piece of sheet music that only contains your part, a monophonic line. Your sheet music tells you nothing about the chord progression or the arrangement. The conductor of course has an orchestral score which contains all the parts of all the instruments laid out in parallel in a system. If you study the conductor's score, you can learn how to read the chord progression and see how the individual parts relate to each other.
I'm a chorister, and in almost all choral music, each member of the choir, along with the keyboard accompanist and the director, read from the exact same piece of music, which is like a conductor's score. Everybody reads from a score that has all the musical parts in a system, including all four vocal parts together with a piano or organ accompaniment, or perhaps a piano reduction of an entire orchestral accompaniment. Choral singers therefore get used to looking ahead and reading not only their own part but also everybody else's part. If they want to develop the facility, choral singers have the opportunity to analyze the chords and harmony, and to look for cues in other parts so they know how their next entrance fits in with the entire arrangement.
Once I tried to play in a classical guitar trio, and they gave me only one instrumental part in the sheet music. I just could not adapt to having only one part to read and not being able to see everybody else's part on my score at the same time. That would require a different skill set, one that I never acquired.
If you, as a trombone player, want to learn about the harmonic progression and the structure of a piece of orchestral music, then get a recording of the entire piece, borrow a conductor's score, and sit down and follow the conductor's score while you listen to the recording. Later on, sit down at a piano with the conductor's score and pick out the chords assembled from the notes of the individual instrumental parts, or analyze them on paper. It's a skill that takes quite a bit of study and practice.
