To understand what the memo is talking about, you have to trace back three separate threads.
Carter Page. In 2013, a number of Russian government agents tried to recruit him, via a lucrative deal with Gazprom. In return, Page admits to feeding them documents (which he claims were “basic immaterial information and publicly available research documents”). As luck would have it, at least one of those agents was already under surveillance (which would later lead to a conviction for espionage), so the FBI asked for and got a FISA warrant to ensure Page wasn’t part of a Russian spy network. That would have been scary, as Page was an advisor to then-candidate Donald Trump. These warrants expire every 90 days, and if the government wants them renewed they have to plead their case to a judge in a special court. The warrant against Page was renewed at least four times, by multiple people and multiple judges.
George Papadopoulos. According to the New York Times, “During a night of heavy drinking at an upscale London bar in May 2016, George Papadopoulos, a young foreign policy adviser to the Trump campaign, made a startling revelation to Australia’s top diplomat in Britain: Russia had political dirt on Hillary Clinton.” By “dirt,” he meant “stolen emails;” a spear-phishing campaign against the DNCC by a Russian government hacking group dubbed “Fancy Bear” succeeded in mid-March 2016, and on April 26th a Russian intelligence agent was teasing Papadopoulos with the emails. When they were released to the public in July 2016, the FBI opened an investigation into Papadopoulos. According to his guilty plea to the Special Investigator, Papadopoulos was bullish on getting Trump to meet Vladimir Putin and kept up his contacts with Russian spies during his time on the Trump campaign. As late as December 4th of 2016, Papadopoulos was publicly calling himself a Trump advisor.
Fusion GPS. In multiple congressional testimonies, Glenn Simpson said that his company was hired by The Washington Free Beacon to dig up dirt on Trump. Fusion GPS’s research found a tonne of it, most notably potential money laundering for Russian oligarchs. Simpson hired Christopher Steele, a former British spy with decades of Russia experience and a solid track record, to investigate the Russian angle. Steele’s work would eventually become “the Trump Dossier” (which may need to be renamed, as I somehow missed a second dossier). Partway through generating the 17 separate memos which would become The Dossier, Trump was crowned the official Republican presidential candidate and the Washington Free Beacon stopped funding it; the Democratic National Committee then picked up the tab. What Steele found had him so terrified that he, with Simpson’s approval, started sharing the information around. The first contact was an FBI agent in Rome; when the FBI themselves seemed oddly disinterested, he went to journalist David Corn and later Senator John McCain. The Dossier’s veracity has improved over time, and the Kremlin may have iced one of Steele’s contacts.
You’ll need a bit more background to fully grok it, but I can sprinkle that in while covering the claims of the memo itself.