Movable antennas (MAs) have emerged as a promising technology to improve wireless communication and sensing performance towards sixth-generation (6G) networks through flexible antenna movement. In this paper, we propose a novel wireless sensing system based on MA arrays to enhance multi-target spatial angle estimation performance. We begin by characterizing the Cramér-Rao bound (CRB) matrix for multi-target angle of arrival (AoA) estimation as a function of the antenna's positions in MA arrays, thereby establishing a theoretical foundation for antenna position optimization. Then, aiming at improving the sensing coverage performance, we formulate an optimization problem to minimize the expectation of the trace of the CRB matrix over random target angles subject to a given distribution by optimizing the antennas' positions. To tackle the formulated challenging optimization problem, the Monte Carlo method is employed to approximate the intractable objective function, and a swarm-based gradient descent algorithm is subsequently proposed to address the approximated problem. In addition, a lower-bound on the sum of CRBs for multi-target AoA estimation is derived. Numerical results demonstrate that the proposed MA-based design achieves superior sensing performance compared to conventional systems using fixed-position antenna (FPA) arrays and single-target-oriented MA arrays, in terms of decreasing both CRB and the actual AoA estimation mean square error (MSE). Fundamentally, the designed MA array geometry exhibits low correlation and high effective power of sensitivity vectors for multi-target sensing in the angular domain, leading to significant CRB performance improvement. The resultant low correlation of steering vectors over multiple targets' directions further helps mitigate angle estimation ambiguity and thus enhances MSE performance.