How to read it
Every note the chart covers is on the staff, lowest to highest, with its fingering underneath. Filled circles are the valves to press, 1st valve on the left; all open circles means no valves. Smaller fingerings underneath are alternates. Notes are written the way they appear in your part, not at concert pitch.
Notes with two names
A♯ and B♭ are the same note, with the same fingering. The chart writes the flat on the staff and the sharp name under it, so you can find either one.
When to use an alternate
Alternate fingerings can help a note that sits out of tune or make a fast passage easier. Notes fingered 1-3 and 1-2-3 tend to run sharp: adjust with your tuning slides or your ear.