A multi-frequency, high-performance, ultrathin composite absorber covering the S, C, X, and Ku bands in the operating frequency range is designed by using a double-layer fractal frequency-selective surface (FSS) pattern composite, whose unit structure consists of the FSS, an intermediate matching layer, and a grounded metal. The finite time domain difference algorithm is used to simulate the structure numerically. The simulated results show that the absorption rate is as high as 90% on average in the six frequency bands of 3.6 GHz, 4.6 GHz, 9.5 GHz, 11.34 GHz, 13.91 GHz, and 16.1 GHz under vertical incidence, which is in good agreement with the experimental results. Due to the symmetrical design of the structure, the absorber is polarization-insensitive. The position of the absorption peak can be effectively regulated according to the dimensions of the cross and square fractals in the cell structure. Moreover, absorption characteristics of the proposed absorber are verified using an equivalent circuit model, and the physical mechanism of electromagnetic wave energy attenuation is investigated by analyzing the surface current, electromagnetic field, and power loss density. The proposed circuit analog absorber could be applied in anti-electromagnetic interference and stealth technology.