This sounds like a multiple choice, but if it isn't I'll just say what I think.
"It gives me pleasure" indicates that he is delighted to announce the news to Congress of the removal of the Native Americans. If we were to take away that section from the passage, it would read something like this:
"The benevolent policy of the Government, steadily pursued for nearly thirty years, in relation to the removal of the Indians beyond the white settlements is approaching to a happy consummation."
Now this sounds cheerful enough, because of the words benevolent (charitable) and happy, but the use of the phrase "gives me pleasure" really sets the tone for the paragraph, and let's you know right away how Jackson is feeling about his announcement.
