Orthogonal time frequency space (OTFS) modulation has been advocated as a promising waveform for achieving integrated sensing and communication (ISAC) due to its superiority in high-mobility adaptability and spectral efficiency. Despite its advantages, due to the complex input-output relation of the OTFS processing, the current OTFS based ISAC schemes require high computation complexity/latency to achieve satisfactory sensing accuracy. In this paper, a concisely fast/slow-time representation for the OTFS input-output relation is derived, which can clearly reveal the physical interactions between the OTFS signals and sensing channels. Under this basis, a novel delay-Doppler spectrum matching assisted active sensing framework is proposed for the OTFS based ISAC systems. The framework consists of three sequentially operated modules, i.e., a coarse searching and triangular spectrum matching module, a local fine tune and quadratic spectrum matching module, and a constant false alarm rate (CFAR) target detection module. The three modules work coordinately to realize joint target detection and high-accuracy target range/velocity estimation. Simulation results show that the proposed framework achieves better detection performance and range/velocity estimation accuracy with much less computations when compared with the reference schemes.