Here is what I have determined so far: Main factors connected to Amin's behaviour that influenced the decision
Amin wanted to decrease his dependency from the USSR and was keen on pursuing a more balanced foreign policy by improving relations with other nations (US, Iran, Pakistan…) → USSR under no circumstances wanted to lose control of Afghanistan, a country they had financially supported from the start. They feared he would do a Sadat. (US or other western force on Soviet border… Military intelligence bases…)
Amin completely ignored Soviet advice regarding Afghanistan’s internal and external politics (they wanted him and Taraki to broaden their support base, and Amin to share his power with his rivals, pursue more moderate policies…)
Amin completely consolidated his power and strategically eliminated his rivals one by one. Taraki’s murder was a turning point for the Soviet Afghan policy.
Amin’s personality is described as «duplicit, brutal...»
Amin's behavior was the major cause of intervention - he killed Nur Mohammed Taraki and lied about it to Brezhnev.
Brief recap:
Nur Mohammed Taraki, first leader of Saur Revolution, has close rapport with Brezhnev. Being a writer, he was rather hapless at ruling, made his own share of mistakes - like premature and half-baked land reform, personality cult, but at least he wasn't as repressive as Amin. He was hopelessly naive, telling Kryuchkov, that what USSR did in 60 years, Afghanistan will do in 5 years.
Amin called Taraki "my father and teacher", Taraki called Amin "my son and best pupil". Amin quickly rose to 2-nd-in-command under Taraki and became indispensable power broker between factions and tribes. Taraki completely trusted Amin in day-to-day operations.
When Amin took over in a coup at Sep 10th, Brezhnev promised to Taraki in face-to-face meeting (Taraki was stopping at Moscow on his way from Havana to Kabul), that USSR guarantees Taraki's personal safety. He also suggested to Taraki to stay in Moscow, but Taraki went back to Kabul.
Amin was getting increasingly brutal and nationalistic - he ordered Khazara bombing raid, alienated tajiks and uzbeks from North by imposing administration of ethnic pashtuns in the North etc. Also, many army officers, loyal to Taraki, were repressed. His attempt to diminish mullahs influence among peasantry was ham-fisted and backfired. Such policies were leading to wide discontent and would clearly end in popular uprising.
USSR leadership concluded, that Amin has to be removed or Afghanistan will blow up really soon. Not all operatives supported this assessment. Chief Military Advisor Lt. Gen. Lev Gorelov, who closely worked with Amin, claims that Amin could have been reigned in and it would have worked better - see [1]. Gorelov was reassigned away from Afghanistan in November, 1979.
Two covert attempts to remove Amin from power have failed. First - to poison Amin came in very close, but Amin was saved by soviet doctor, who wasn't informed about attempt on Amin's life. Second attempt was by PDPA members - "Gang of Four", but Amin's secretary/bodyguard Taroon took the bullet.
Amin has ordered Taraki murder (who was smothered by pillow in the Pul-e-Charkhi prison) and tried to cover up by telling that Taraki "is not feeling well".
Some sources have claimed (see [3]), that it was Brezhnev ambiguous reply hastened Taraki's death ("it's your choice what to do with Taraki"), but it there's no confirmation in primary sources, that Brezhnev has ever communicated with Amin, moreover, there are many mentions that Brezhnev was carefully avoiding Amin and any signs of official endorsement.
In fact, USSR made an inquiry about Taraki's health and Amin promised to personally monitor Taraki's health and provide all necessary means for his recovery. At that moment Taraki was dead and Amin knew it.
When Brezhnev was told about Taraki's murder, he became really upset and later insisted on direct military intervention. Also, KGB leadership supported intervention against Ministry of Defense and General Staff recommendations, although by 1983 Andropov (he was KGB Chairman in 1979) recognized it was a mistake. KGB preferred Babrak Karmal as more pliable vs both Taraki and Amin.
Acad. E. Chazov, leading USSR cardiologist, served as a "Kremlin's doctor" at a time and was quite close to Brezhnev. By 1979 Brezhnev has very serious health issues and saw Chazov almost daily.
Here's quote from Chazov's memoirs (Brezhnev very upset about Taraki's death, that his promise to Taraki about safety was broken and calls Amin 'scum') :
(russian)
Брежнев, несмотря на снижение способности критического восприятия, бурно переживал это событие. Больше всего его возмущал тот факт, что только 10 сентября, незадолго до этих событий, он принимал Тараки, обещал ему помощь и поддержку, заверял, что Советский Союз полностью ему доверяет. «Какой же это подонок — Амин: задушить человека, с которым вместе участвовал в революции. Кто же стоит во главе афганской революции? — говорил он при встрече. — И что скажут в других странах? Разве можно верить слову Брежнева, если все его заверения в поддержке и защите остаются словами?»
Sources
russian - Lev Gorelov. How it happened - interview with Lev Gorelov by Artyom Sheinin.
russian - Excerpt from Protocol 172 - Oct 31, 1979 - Detailed description of Amin's missteps. Politburo has decided to plan for Amin removal.
Misdaq, Nabi (2006). Afghanistan: Political Frailty and External Interference. Taylor & Francis. p. 125. ISBN 978-0415702058. - oft-cited secondary source with many factual mistakes.
russian - Е. Чазов. Здоровье и Власть : Мемуары "Кремлёвского врача" = E. Chazov. Health and Power: Memoirs of the 'Kremlin Doctor'. Moscow: Novosti, 1992 - Chazov's memoirs wasn't translated into English, AFAIK.
English - The Origins of the Soviet-Afghan War - Revelations from the Soviet Archives - some Politburo protocols about Afghanistan translated to English (not all, though)