Context. The properties of variable stars can give independent constraints on the star formation history of the host galaxy, by determining the age and metallicity of the parent population. Aims: We investigate the pulsation properties of 84 Anomalous Cepheids (ACs) detected by the OGLE-III survey in the Large Magellanic Cloud (LMC), in order to understand the formation mechanism and the characteristics of the parent population they came from. Methods: We used an updated theoretical pulsation scenario to derive the mass and the pulsation mode of each AC in the sample. We also used a Kolmogorov-Smirnov test to analyze the spatial distribution of the ACs, in comparison with that of other groups of variable stars, and connect their properties with the star formation history of the LMC. Results: We find that the mean mass of ACs is 1.2 ± 0.2 M&sun;. We show that ACs do not follow the same spatial distribution of classical Cepheids. This and the difference in their period-luminosity relations further support the hypothesis that ACs are not the extension to low luminosity of classical Cepheids. The spatial distribution of ACs is also different from that of bona-fide tracers of the old population, such as RR Lyrae stars and population II Cepheids. We therefore suggest that the majority of ACs in the LMC are made of intermediate-age (1-6 Gyr), metal-poor single stars. Finally, we investigate the relation between the frequency of ACs and the luminosity of the host galaxy, disclosing that purely old systems follow a very tight relation and that galaxies with strong intermediate-age and young star formation tend to have an excess of ACs, in agreement with their hosting ACs formed via both single and binary star channels. Based on observations released by the Optical Gravitational Lensing Experiment (OGLE-III).