Some version of this question is asked relatively often, particularly with regard to Nazism; see the "How Socialist was National Socialism?" section of the FAQ.

To give you a quick answer: the Nazi Party was kind of an ideological hodgepodge in its early years, drawing support from both the left and right fringe, including some people whose economic views can be considered quite close to Communism. After he became leader, Hitler had his own far-right ideas of where he wanted to steer the party, but the party was still so weak that he was happy to court anyone who might lend him support. As I understand it, the primary unifying ideological factors of the early NSDAP (Nazis) were antisemitism and nationalism, both of which stood in stark opposition to the German Communist (KPD) articulation of internationalist and comparatively egalitarian Leninism. Certainly the fact that Marx was a Jew also helped to inflame the Nazis' anticommunist rhetoric; at some point the Nazis merged antisemitism and anticommunism into one grand conspiracy theory called Judaeo-Bolshevism. The perceived link between the two revolved around the idea that Bolshevism, whose explicit aim was a worldwide socialist dictatorship transitioning into a stateless communist society, was a tool invented by the Jews to enact their nefarious plots of world domination as outlined in the hoax document Protocols of the Elders of Zion. This conflation was everything the Nazis hated, being both Jewish and antinationalistic—despite any potential similarities between the NSDAP and KPD economic plans.

In some sense, because the early NSDAP and the KPD shared certain views and were so completely opposed on others, they inevitably ended up competing for left-wing voters, with the NSDAP trying to convince the German voter that his greatest enemy was the Jew (and, later, the communist) while the KPD claimed his greatest enemy was the bourgeois capitalist. The two parties fought many ugly street battles before Hitler became chancellor in 1933 and promptly banned the KPD. The left wing of the NSDAP then began to agitate for concessions to their economic agenda, which Hitler's right-wing supporters (drawn largely from the army and the wealthy patricians) could not abide. This is getting out of the scope of your question, but for more on what happened next, see Night of the Long Knives.

I'm afraid I don't know as much about anticommunism in fascist Italy, but I imagine it also has something to do with Italian communists failing to be sufficiently nationalistic for Mussolini's taste.

The foundations of marxism and fascism are completely opposed to one another. In the 20th century marxism was the dominant current in socialism and communism, even though they aren't synonymous.

Marxism is based in class struggle, the idea that different groups in society are fundamentally opposed to one another because of their economic interests. Communists want to put these interests against each other in revolutionary struggle until the proletariat comes out on top. Any attempts to unite the different classes in society (against another ethnic or racial group, for example) is therefore an attack on the proletariat, as it keeps the struggle down and maintains the status quo.

On the opposite side, fascism is based in nationalism, and national unity. To them, the struggle is between different racial, ethnic or national groups, and any attempt at starting a class war divides the nation and weakens it.

So, yeah. The two were never going to be friends.