Multimodal sarcasm detection (MSD) aims to identify sarcasm polarity through diverse modalities (i.e., image-text pairs), which gains increasing attention. While significant advancements have been witnessed, the existing approaches still face two major issues: lack of explainability and weak generalizability. In this paper, we introduce a new large vision-language model (LVLM) dubbed S³-MSD for explainable and generalizable MSD through three key components. For explainability, we develop (1) a self-training paradigm bootstrapping answers with explanations automatically, and (2) a self-calibrating mechanism rectifying flawed explanations. For generalizability, we design (3) a self-focusing module amplifying visual semantic entities through preference optimization, to mitigate textual over-reliance. Experimental results on both in-distribution and out-of-distribution (OOD) benchmarks demonstrate that S³-MSD consistently outperforms state-of-the-art methods in detection performance. Furthermore, the proposed S³-MSD provides persuasive explanations, as validated by quantitative and human evaluations.