The origins of this phrase appear lost to time, but it was first heard when people noticed that the Left had started moseying over toward full-blown Communism after the American Civil War.
Abe Lincoln showed us what happens when conservatism goes bad. He identified with his humble origins to the point of being a Marxist-style class warfare expert who was willing to destroy his own people for the symbolic victory of racial equality.
The Left of the day delighted in him of course. After Irish immigration, American culture dissolved, and the only way to keep the remaining people united was to find something for them to hate. They figured slavery was indenfesible, so why not fixate and agitate in that direction?
This worked really well for its original purpose, namely to unify a voting bloc and social group, but it got out of hand, sort of like Hitler’s scapegoating of Jews, and soon the Northern states were pushing the country toward war.
Lincoln cobbled together some law, called it civil rights, and used this as the basis of uniting the country, apparently ignoring the fact that this converted us from a government that supports a people into a government that seeks to create total ideological compliance to “equality.”
Guess those lawyers who came from backgrounds other than humble origins might have known something that Abe missed.
In any case, at the moment that Abe devoted our nation to the cause of keeping itself unified, and chose equality as the means of doing that, he pointed our Leftists toward full Communism, a belief system which surmises that government must be more than a caretaker; it must shape its population toward an ideological correct behavior.
After that point, the Left — which at that time, was the Progressive Republicans — recognized that it had one course to follow. It needed a socialist style total state to enforce equality, since anything else would collapse quickly, with equality for outsiders like Africans used as a tool to remove any barriers of culture, family, faith, economics, and sanity that stood in its way.
Add to this the basic fact that all Leftists aspire to the same thing, since all are egalitarians, and you can how all roads lead to Full Communism, just at different speeds. Egalitarians believe that the purpose of civilization is to make people equal, and that this alone can be its highest achievement and singular morality. Anything else is, in their view, evil.
As you can see, a Democrat wants to use gentler methods than a Communist, but they agree on the basic idea: a diverse total state run by bureaucrats for the purpose of equality. The Democrat wants high taxes, not the Stasi, but he will take that if he needs to. After all, it serves his goal.
The phrase “A Communist is just a Democrat in a hurry” refers to how all roads lead to Rome, in this case, equality; for that reason, all egalitarians are the same, and differ only by degree. Communists, Democrats, Socialists, Anarchists, and libertarians all want the same basic model of civilization.
Right-wingers on the other hand aspire to an Invisible Order compromised of a transcendental assessment of natural and divine order. This is nowhere near as simple as “make equal,” so it suffers from being less popular, but people return to it every generation as the Leftist plans implode, as they always seem to do.