Free-text rationales justify model decisions in natural language and thus become likable and accessible among approaches to explanation across many tasks. However, their effectiveness can be hindered by misinterpretation and hallucination. As a perturbation test, we investigate how large language models (LLMs) perform the task of natural language explanation (NLE) under the effects of readability level control, i.e., being prompted for a rationale targeting a specific expertise level, such as sixth grade or college. We find that explanations are adaptable to such instruction, but the requested readability is often misaligned with the measured text complexity according to traditional readability metrics. Furthermore, the quality assessment shows that LLMs' ratings of rationales across text complexity exhibit a similar pattern of preference as observed in natural language generation (NLG). Finally, our human evaluation suggests a generally satisfactory impression on rationales at all readability levels, with high-school-level readability being most commonly perceived and favored.