The logic that you learn in school is called formal logic because it is only concerned with form, not content. If you start from bullshit premises, you can logically derive bullshit conclusions. Hegel, being a turbo idealist, thought this was wack and wanted a logic that was concerned with content.
His grand plan was that if he started with the most basic concept and thoroughly investigated it, he must necessarily uncover all the secrets of the world up to the Absolute Spirit, which is like God and contains every concept ever and governs the rules of the universe. Of course it being a logic if you did the same you must necessarily reach the same conclusions and if you did not, it must be because you are stupid.
His method is the following. First you take a concept and try to give it a definition ("thesis"). You thoroughly study this definition and you realize that you actually defined something entirely different than what you wanted, something like the opposite (negation/"antithesis") of your original concept. Not being a loser to give up, you start investigating this new concept, trying to define it, only to realize that upon further investigation, this second concept now really looks like what you originally wanted. So you go back to defining it, only to realize that you are back at the negation... Until you take a step back, pull a new concept (sublation/"synthesis") out of your ass to distinguish the two previous cases, by both preserving them and abolishing them. Then you start defining it only to realize that it is actually something else... and you repeat until you have the ultimate concept that puts everything in place. Of course, since he has to explain everything, he does not strictly follow his own method.
As an example, consider (pure) Being, which is the concept of something existing. Without anything specific, just the fact that something is. Hegel tells you that investigated on its own, Being is actually a lot like Nothing, the concept of not existing. And if something is not, then it actually is, because we are talking about concepts and not object, right? To solve this contradiction, he introduces Becoming, which is this passing from Nothing to Being.
An example closer to Marxists can be found when examining the interests of the worker and the boss. The boss' interest is to pay as low wages as it can for as much labour as can be extracted, while the worker's interest is to work as little as possible while taking home as much as possible. If their interests are so opposed, how come they end up cooperating always? It is because a third concept, the company's interest. It preserves the opposition, the worker and the boss still struggle to win their little victories in dividing the profit, but it also solves it as they can only do this in the context of the company's interest, both of their survival depend on the continued commercial success of the enterprise. This is an appealing picture, but only an imbecile would claim that companies were invented to contain and neutralize class struggle.
Yet when Marxists talk about the "dialectic" being "in motion", this is what they are talking about. They seriously believe that history is governed by some inhuman forces that evolve on their own. The fact is, Hegel would have been long forgotten if not for the Marxists insistence on preserving his delusions. They think Hegel is somehow relevant to understanding Marx because the first chapter of Capital is structured like a parody of Hegel and makes use of Hegel's terms. As a joke, only to end up with basically saying "this is how the bourgeois thinks lmao they are stupid fucks who thinks commodities move on their own". Of course if Marxists were capable of thinking for themselves they would be anarchists by now.
Oh, and when they accuse you of "not being dialectical", as if that was something to be ashamed of and not a sign of not being deluded, it means that they are upset that you have pointed out their incoherence.