It's a bit difficult to do this without reference to a specific theory or without further specification so this is going to be very general. I also don't know how far you cast your question so I am going to start with basics and I hope you can forgive me if you know most of this already.

Basically, in most cases a work in history whether it is an essay or a longer work is structured around a central question, meaning the thing you want to find out. Designing such a research question is an essential step in the work of a historian because it already is a lot of work. It's based on work within the historiography, meaning it is designed to find out something we don't know yet ideally (this is not such a strict requirement in the case of essays though it can be) and often also on the source material you know is available (finding that out is also a piece of work mostly). This question can take the form of f.ex. "What were the influential factors for the escalation of violence by German troops in the Soviet Union?" or "What role did Protestant teachings play in the German peasant wars?" – basically something you want to know.

To answer your question you need a methodology, meaning a way to come from your question to an answer with your source material. Say, for instance, you decide to ask about the influential factors for the escalation of violence by German troops in the Soviet Union you can go about answering that question in various ways: You can, say, focus on analyzing propaganda aimed at the troops and the images of the enemy it presents. Or you can, for example, focus more strongly on the situational circumstances of individual units and see if there is a connection between the escalation of violence and the experience of concrete or imagined imminent danger. Both of these are viable but they necessitate different methodological approaches: To focus on propaganda and the images of the enemy would work with a discourse analysis, concretely, to analyze what is said and thought on a social level about real or imagined enemies. To go the situational route would, on the other hand, need a methodology tailored to explaining what makes people act in a certain way such as the sociological theories of Erving Goffman.

Additionally, historical work necessitates to formulate a (hypo)thesis, meaning a potential answer to your research question you set out to prove with your methodology. Staying with the same example, you hypothesis can be that it was the general Antisemitism and anti-Communism of Nazi German society that lead to the escalation of violence in the USSR or you can formulate the thesis that it was the concrete experience of Partisan warfare that weighed more heavily as a factor in said escalation.

So, to quickly recap: A historical work is based on a research question, formulates a (hypo)thesis as to the answer of said question and has a methodology it utilizes in answering said question and (dis)proving the (hypo)thesis. Every one of these steps can be (and mostly is) based on theory. A theory delivers you a priori and basic assumptions about how the world/society/individuals work. That's what makes it a theory, as a system of general thought about society and the world. If you, say, embrace the theory of classical Marxism, you embrace the thinking that culture and social ideology is mostly based on the economic base and material conditions. If you do so, you will naturally ask different questions about the past, formulate different (hypo)theses about the questions, and use a different methodology than, for instance, someone who embrace Foucault and discourse theory. So already what questions you ask and what theses you formulate about potential answers is something influenced by theory.

However, where theory is most important is regarding the methodology. Methodology is not exactly theory but most methodology is based on theory because theory delivers the tools of methodology. To discuss this once again with the above example: You want to show that the German escalation of violence in the USSR is based upon the experience of Partisan warfare. How can you show that? You need, for example, a methodology that can go into detail about how people act. Such a methodology will be based upon one or more theories that lay out in general how people come to act in general. You could go with the above mentioned Erwing Goffman: Goffman says humans act because of three conceptual frames, meaning ways to organize experience about the world: A social frame, a situational frame, and individual frame. A social frame is how a society culturally determines the organization or experience; a situational frame is how we can perceive an individual to perceive certain situations (a threat etc.); the individual frame is based on how an individual's previous experience and upbringing determines how they organize information (whether they were raised Catholic or Protestant, whether they were veterans etc. etc.).

Based on such a theory, you can formulate a methodology to answer such a question. You can, for instance, look at sources from selected examples of individual soldiers and try to determine thier individual frame of reference to learn something larger about the dynamic of escalation. Or you can focus on what the Wehrmacht taught about situations in order to arrive at the situational frame of reference. Or, should you instead focus more on the propaganda and go the other route, you'd need, say, a discourse analysis that takes what words and concepts are frequently invoked in propaganda in order to determine what the social disocurse about Partisans, Jews etc. was.

So, in short, theory is indispensable to formulate methodology but also influences question and thesis. In a university essay context, I'd wager that the part about methodology is the most important one and hence this is where I'd see theory as most important for you.

Do they specifically mention theory or do they say historiography? If the latter then it's possible that they're referring to the more specific meaning of the term. While historiography does mean theory in general, when people talk about the historiography of a certain subject they generally mean the arguments surrounding it, for example intentionalism vs functionalism in the Holocaust or the Sonderweg debate. It's possible that your university places a lot of weight on critically engaging with and showing awareness of the relevant arguments and debates of the topic.