What information is the astronomer presenting at the beginning of the poem?

It is unclear what specific information is being presented by the astronomer at the beginning of the poem, but the reader can gain a better idea via the context that Walt Whitman provides.  Firstly, the astronomer is "learn'd," meaning educated.  This is a person who is either a teacher or a speaker, giving some sort of presentation regarding the cosmos.  We are told that he "lectured with/ much applause in the lecture room" (5-6), which suggests that he is a popular speaker, able to draw a large crowd.  Further, during the lecture, the narrator observes nearly all manner of data,

When the proofs, the figures, were ranged in columns before me;/When I was shown the charts and the diagrams, to add, divide,/and measure them; (2-4)

The lecture is filled with many visual aids; one can assume that the charts and diagrams display the astronomer's findings.  The fact that mathematical functions such as addition and division are used in conjunction with the charts and diagrams suggests a great amount of research, and also suggests that the astronomer is explaining the steps in his method as he lectures. 

Listening to the astronomer lecture on about the charts, numbers, and statistics makes the narrator feel "tired and sick" (7) until s/he finds comfort by going outside into the night and looking up "in perfect silence at the stars" (10).  The view of the night sky, in all its natural splendor, renews the narrator.  This allows the reader to make the educated guess that the astronomer was lecturing on some new findings regarding the distant cosmos itself, in an effort to further explain the workings of the galaxy. For the narrator, this explanation is unwelcome because it takes away the wonder and amazement of the universe.