What was going on in the world which made the Texas vs Johnson case important at that point in history?
To me, this case was important at the time simply because it was an extension of the culture wars that had been going on since the time of the hippies and the Vietnam War. It represented yet another battle between liberals and conservatives.
In the late 1980s, many of the people who were becoming politically important were of the "Vietnam Generation." These were people for whom attitudes towards the country and patriotism were important dividing lines. They continued to remember the ways in which these issues had divided people during the Vietnam War era and, in some ways, they seemed to want to refight those battles.
In that context, this case was explosive. It pitted the conservatives (who were in ascendance due to the Reagan presidency) against the liberals who were still opposed to traditional views of patriotism because of the way that those were (to them) abused by the powers that be during the Vietnam Era.
So I think that the main thing that was going on was this budding culture war between liberals and conservatives that continues to this day.