Good question, friend!

First, a brief bit of background. From what I understand, medical consensus is that HIV started out as a disease among chimpanzess called Simian Immunodeficiency Virus, and spread to humans in the region called the Congo, who sometimes ate chimpanzees. From there, an individual or individuals who immigrated to Haiti allowed the disease to cross the Atlantic with them. And someone or someones who moved from Haiti to the U.S brought it to the North American continent, with the first reported case (though they didn't known it was AIDS back then) surfacing in 1981. (1)

So, why the 80s? A few reasons. I know good answers cite many sources, but Dudley Clendinen's Out for Good: The Struggle to Build a Gay Rights Movement in America is so comprehensive it should suffice :)

According to Clendinen, during the 70s the American gay male community was extremely uninhibited sexually. Partly as a reaction to the sexual repression of the Cold War (gays could barely even enjoy bars because of the common misconception they were easily "seduced" by Communists), gay men created for themselves an extremely libertine culture in the late 60s and 70s. Drugs were everywhere, and there are stories of men having a dozen sexual partners in a single night at some bars. Naturally, venereal diseases ran rampant in this environment. So when that unfortunate Haitian immigrant(s) came to the U.S (sometime in 1968, according to the New Scientist), conditions were ripe indeed for the spread of a terrible and virulent new epidemic.

However, the nature of the illness--which attacked the immune system and left you very vulnerable to other diseases--made identifying it quite difficult. So even though it had been spreading before the 1980s, AIDS was only identified when doctors noticed that otherwise very rare diseases (like Kaposi's Sarcoma) were showing up among gay men, realizing that those diseases usually attacked people with weakened immune systems, and connected the dots to figure out some force was weakening many people's immune systems. They naturally called the affliction "Autoimmune Deficiency Syndrome" and the strange, previously unknown virus that caused it the "Human Immunodeficiency Virus."

Thus, the AIDS epidemic was brought to light in the 1980s--this was not the decade it "began," necessarily ('patient zero' came to the U.S earlier) but was the decade when it was both identified and when it reached truly massive proportions, due to being spread so widely by the promiscuity of the previous years.

Endnotes:

1: Neil Bowdler, "Key HIV strain 'came from Haiti" for the BBC News, last accessed at http://news.bbc.co.uk/2/hi/health/7068574.stm on Nov. 19, 2015, and "Timeline: HIV and AIDS" from New Scientist, last accessed at https://www.newscientist.com/article/dn9949-timeline-hiv-and-aids#.U-UHyfldV8E on Nov. 19, 2015.

2:Dudley Clendinen and Adam Nagourney, Out for Good: The Struggle to Build a Gay Rights Movement in America (Simon and Schuster, 1999), pp. 448-490.