This is an active topic of debate among professional philosophers of science today. Julian Reiss explains the problem nicely in his paper "The Explanation Paradox". His paper is focused on economics, but we can generalize the problem. Basically, the following statements are all highly plausible:
- Explanations must be true.
- Many fields of science make heavy use of simplifications, idealizations, and false assumptions in their research.
- These fields produce explanations.
But these 3 statements are inconsistent, so at least one must be false. Some professional philosophers think we should stick with #1; they are called "factivists," because they think explanation requires "facts," i.e., true claims. Factivists either reject #2 (these simplifications and idealizations don't play an "essential" role in the explanations) or #3 (Reiss gives the example of Anna Alexandrova and a coauthor, who simply think that economics doesn't produce explanations).
Non-factivists think that explanations don't require truth. Often — though not always — they argue that understanding is a subjective feeling or qualitative grasp of the phenomena. A false explanation can produce this feeling or qualitative grasp, perhaps even better than a true-but-non-explanatory description.
Two recent books worth mentioning are Angela Potochnik's Idealization and the Aims of Science and Kareem Khalifa's Understanding, Explanation, and Scientific Knowledge.
Potochnik is a non-factivist, so she rejects #1. On her view, the primary epistemic aim of science is understanding rather than truth, and idealizations actually promote this aim (because they allow us to cognitively grasp the phenomena). But she also argues that the causal patterns described by an explanation must be "embodied" in the phenomena, perhaps only "imperfectly." So, despite being a non-factivist, she still thinks explanations require some kind of representational relationship to the phenomena. (Unfortunately, I had trouble understanding how this is supposed to work in complex cases with "imperfect embodiment.")
Khalifa is a factivist (or, in his terminology, a "quasi-factivist"). But he argues that the truth requirement only applies to aspects of explanations that we believe to be true. We can adopt a different attitude, called "acceptance," towards the idealizations. We don't believe the false assumptions are literally true; but we accept them as convenient fictions, useful for simplifying calculations or bracketing complications (perhaps only temporarily).
I mention these two books because, while they adopt formally different labels, their views on truth and explanation are very similar. Both think that explanations must be "true-like," at least to some degree; but both also allow false elements to play a role in explanation.