In this paper, we study the level set estimation of a spatial-temporally correlated random field by using a small number of spatially distributed sensors. The level sets of a random field are defined as regions where data values exceed a certain threshold. We propose a new active sparse sensing and inference scheme, which can accurately extract level sets in a large random field with a small number of sensors strategically and sparsely placed in the random field. In the proposed active sparse sensing scheme, a central controller dynamically selects a small number of sensing locations according to the information revealed from past measurements, with the objective to minimize the expected level set estimation errors. The expected estimation error is explicitly expressed as a function of the sensing locations, and the results are used to formulate optimal and sub-optimal selection of sensing locations. Simulation results demonstrate that the proposed algorithms can achieve significant performance gains over baseline passive sensing algorithms that do not proactively select the sensing locations.