"At last, he was being given a chance to participate in the actual workings of government itself" - Is it the passive voice of the past continuous? Why it's being used? Can one say - "...he was given..."?
Why is the passive voice of the past continuous being used? (Why is it being used? -- Note the inversion of "is" and "it" for the questions form.) Can one say - "...he was given..."?
I will answer your second question first. Yes, one can say that. The meaning is slightly different. Let's contrast two versions of the sentence, with the continuous and without it. I will give this poor guy a name, to make it easier to talk about him.
A Without: "At last, Bob was given a chance to participate in the actual workings of government itself."
B With: "At last, Bob was being given a chance to participate in the actual workings of government itself."
In A, we learn about something that happened at a specific point in time. We can imagine Bob showing up to work one Monday morning after 6 months of boredom. On that Monday morning someone gave him something meaningful to do.
In B, the narrator is describing how Bob was feeling about his job during a period of time, for example during months 7, 8 and 9 on the job. During this period, we can imagine Bob walking to work every day with a spring in his step, because finally, his boss was giving him meaningful things to do every day.
To answer your first question, the passive voice allows us to express all of this without referring to the person who gave Bob the assignments. The focus is all on Bob.