This is pretty much the question to my MA thesis that I'm working on. I'll give a shorter answer than that.

I contend that it stems back from the 1920s and the establishment of Bolshevik power in Georgia (and Abkhazia). Prior to this (meaning in the Russian Empire; independent Georgia had different interactions with Abkhazia, which I'll note). Soviet policy at the time gave Abkhazia the status of union republic (quickly reduced to "treaty republic," a unique case in the USSR that had it on equal status with Georgia, until 1931 when it was reduced to just an Autonomous SSR), partly because the Bolsheviks had support in Abkhazia after the 1917 revolutions, much unlike Georgia (which was a Menshevik stronghold).

Stanislav Lakoba, the leading Abkhaz Bolshevik, was essentially given free reign, and Abkhazia was seen as his personal fief. This was in part because Lakoba was an ally of Stalin, working with him when Stalin was establishing himself as a young revolutionary in the Caucasus at the turn of the century. It was because of this relationship that Lakoba also managed to prevent collectivization when that began; he effectively traded this in return for reducing Abkhazia's status to an autonomous republic in Georgia. However this proved fatal.

Lavrenty Beria was the First Secretary of the Georgian Communist Party at this time. He was ethnically Mingrelian, which is a group related to (but distinct from) Georgians, and was born in Abkhazia. He didn't like Lakoba's standing regarding Stalin, and worked to usurp that and put himself there. So in December 1936 he had Lakoba invited to an opera in Tbilisi, and poisoned him.

With Lakoba out, Beria was able to put his own guy in charge in Abkhazia, and began a process of "Georgianization." The two major features of this was in terms of language and education, and demographics. The Abkhaz language, which had adopted a Latin-based script in the 1920s, changed to a Georgian script. Georgian and Russian were imposed on the people, even though few of them understood Georgian (Russian less so). Education was effectively ended in Abkhaz, with Georgian and Russian being used instead. The Abkhaz Party saw the proportion of ethnic Georgians rise greatly in this period, far more than the demographics of the region had. As well thousands of people were "voluntarily" resettled in Abkhazia, most of them being ethnic Mingrelians and Georgians. This upset the Abkhaz, as the titular nationality was losing its prominence in the region. In effect Abkhazia was made subordinate to Georgia, something it had not really done before (I noted the Democratic Republic of Georgia above; they had wanted to do the same, but due to the uncertainty of the times, Abkhazia was effectively on its own. The 1921 Georgian Constitution did specify that Abkhazia would have been autonomous, but as that was proclaimed 4 days before the Bolsheviks took over Tbilisi, it was never clarified what this would have meant or done).

Once Stalin, and Beria, died most of this ended (Abkhaz readopted a Cyrillic-based script, for example; one it uses today). But the process had been settled. Protests occurred nearly every decade, like clockwork, in 1957, 1967, 1978, 1989, in Abkhazia against Georgia, and each time they gained some concessions.

But even after 1989 it was relatively calm in Abkhazia. It was only in 1992 that things really got bad. With Gamsakhurdia, a virulent nationalist, in power in Georgia, he upset the Abkhaz by refusing to consider their calls for autonomy (based on the 1925 constitution that made them a treaty republic). So in August 1992 the Georgian military entered Sokhumi after Abkhazia declared independence, starting the first war and all the subsequent events.

This is pretty much a simplified version of what's in my thesis, or will be. Expansion or clarification of anything can be provided as well.