So expect plenty of spurious stories about kidnapped kids, celebrities going wild and whatever else the papers can churn up to fill the gaps. Since nothing of any genuine national importance is published for two months, this sometimes leads to public concern/uproar/panic over articles that would usually be pushed to an inside page and forgotten about within days. In a worst-case scenario, this can lead to a Pædo Hunt, possibly a literal one in the mould of Torches and Pitchforks.
The news media in the USA follow a similar path in July and August, with Shark Attacks! and other filler stories when business in Washington, D.C., and New York wind down for the summer, and the paucity of new scripted programming means television networks and stations need something to draw viewers in to watch. This sort of thing also happens during Sweeps, especially with local news programming. One summer was actually referred to as "The Summer of The Shark" because the rest of the news was so slow, despite the fact that shark attack rates that year were no higher than usual. Really, they just seem to hate sharks. It was also the summer of the Chandra Levy murder, which brought down a congressman who was wrongly accused of said crime because the media didn't check their facts. But once September came along, the 2001 silly season ended with a big thud in a really, really unexpected way.... (Somewhat) fortunately (or very unfortunately) for Americans, elections occur every two years in the United States—with presidential ones coming every four—and August is the buildup to real Campaign Season: Labor Day. This means that politicians tend to start really hitting the campaign trail at that time, giving the newsmedia something better to cover than the eating habits of unusually large fish.
(A side-note for motorsport fans: No, this trope has nothing to do with the period during the season in which drivers and crew members make a bunch of deals with other teams... especially as those never get into the newspaper, even during these periods.)
For the United Kingdom, 2008's Silly Season went on (and on) a bit... longer than scheduled, to the relief of politicians who were spared having to talk about the recession. In the United States, July and August 2009 were infamously dominated by stories surrounding the death of Michael Jackson.
The term's been around since at least 1887; in 1950, the scifi author Cyril M. Kornbluth wrote a story with this trope as a title, in which invading Martians use the phenomenon to catch humanity off-guard and take over the Earth.
This isn't restricted to the mentioned countries: Chile, being in the southern hemisphere, has its own silly season during the summer vacation months of January and February. The Dutch concept of "cucumber time" is similar - a somewhat outdated term for a time in which nothing noteworthy happened, and thus the newspapers would report on, say, the growth of cucumbers (or potatoes, or cabbage ...any usually-ignored crop yield). In Spain, this phenomenon is called "serpiente de verano" ("summer snake"), while the Swedes call it "nyhetstorka" (news-drought).