The overwhelming threat of a centralized authority figure that faced no internal check or limitation was reality under the rule of Hitler and Mussolini. Both leaders were quite "efficient" in eliminating competition and opposition. They also ensured that once they obtained power, it was consolidated and no other force could emerge internally to destabilize them. This ended up representing an end that created the setting for overwhelming threats and positions that caused challenge to all of those who opposed the growing threat that both leaders separately constituted, and then jointly represented when they formed alliances with one another.
Hitler and Mussolini and the other leaders within the Axis Powers during World War II promoted a Fascist form of government, or what we would today consider an extreme "right-wing" form of leadership. Among the many countries that joined the Axis, this was their uniting principle, which arose as a counterweight to Communism, or an extreme "left-wing" form of leadership. The perceived threats of Germany and Italy were the possibility of the spread of Fascism; but the underlying reason why Fascism spread was due to the rapid spread of Communism in the former half of the 20th century throughout the new Soviet Union and other countries within Europe. Part of the reason the US deferred fighting in Europe until 1944 was neither political ideology was acceptable; to attack Fascism was, at the time, to necessarily uphold Communism. With the defeat of Fascism, the 50 year Cold War conflict between Democracy and Communism, between the US and USSR began.
So are you asking what they thought were threats? Or why others thought they were threats? I'm going with the first of these.
Mussolini and Hitler basically thought there were a couple types of threats. They would have worried about threats from within -- people who weren't similar enough to the dominant culture. This is why they were anti-Jew, anti-Gypsy, anti-gay, etc. They wanted their populations to be as homogeneous as possible.
But they also thought that there were threats from other countries. This was mainly a concern for Hitler, who thought that the French and, especially, the Soviet Union were threats.
To answer this questions we need to clarity on the period e are talking about and the people of nation feeling the threat.
For example, perhaps for a long time Hitler and Mussolini felt no threat from each other. While for Jews in Europe it was not just a threat it was a real misery for a long time. For them it was a real calamity causing immense amount of death, destruction and misery.
For many countries, who were part of the axis power there was no threat or perhaps much less threat than, that for allied countries. In general. in a period around 1941 and 1942, most of the threats were actually being executed.
Then turning to the German people who supported Hitler initially, began to see him as a big threat subsequently to personal safety and freedom of people, and in later Hitler Had to resort to very strong repressive methods to remain in power. Many of close associates of Hitler also started viewing Hitler as a danger for the peace and prosperity of Germany.
But speaking in very general term we ca say that bot Hitler and Mussolini represented great threat to established human values such as freedom, equality, and peace.