Why are the propos such a big deal to Katniss and the district citizens?
The propos in Mockingjay were important for multiple reasons: they were propaganda, they were distractions for a rescue mission, and they were entertainment.
As the name suggests, propos were propaganda spots, intended to call attention to the evil of the Capitol. They mostly came from the side of the Rebellion, often focusing on Katniss and her involvement in the Hunger Games and her role as the Mockingjay; but the Capitol did have a few of their own. But other things such as interviews, messages, and advertisements were also a main theme of the propos. These served as a sort of rallying cry for the citizens of the districts, urging them to action against the Capitol.
They used a propo to serve as a distraction when Katniss and others went to rescue Peeta, Johanna, Annie Cresta, and Enobaria. This particular propo featured Katniss talking about how she met Peeta, and also Finnick Odair revealing all the secrets he had learned from Capitol citizens over the years.
Finally, one propo that was simply meant to entertain was the one about Finnick's marriage to Annie Cresta. It brought a lightness, a happiness to those who watched it.