Reconfigurable intelligent surface (RIS) can be employed in a cell-free system to create favorable propagation conditions from base stations (BSs) to users via configurable elements. However, prior works on RIS-aided cell-free system designs mainly rely on the instantaneous channel state information (CSI), which may incur substantial overhead due to extremely high dimensions of estimated channels. To mitigate this issue, a low-complexity algorithm via the two-timescale transmission protocol is proposed in this paper, where the joint beamforming at BSs and RISs is facilitated via alternating optimization framework to maximize the average weighted sum-rate. Specifically, the passive beamformers at RISs are optimized through the statistical CSI, and the transmit beamformers at BSs are based on the instantaneous CSI of effective channels. In this manner, a closed-form expression for the achievable weighted sum-rate is derived, which enables the evaluation of the impact of key parameters on system performance. To gain more insights, a special case without line-of-sight (LoS) components is further investigated, where a power gain on the order of $\mathcal{O}(M)$ is achieved, with $M$ being the BS antennas number. Numerical results validate the tightness of our derived analytical expression and show the fast convergence of the proposed algorithm. Findings illustrate that the performance of the proposed algorithm with two-timescale CSI is comparable to that with instantaneous CSI in low or moderate SNR regime. The impact of key system parameters such as the number of RIS elements, CSI settings and Rician factor is also evaluated. Moreover, the remarkable advantages from the adoption of the cell-free paradigm and the deployment of RISs are demonstrated intuitively.