Stationary velocity-perturbation streaks have recently been identified in laminar swept-wing boundary-layer flow interacting with a surface forward-facing step. Streaky structures at the step promote early laminar-turbulent transition under certain conditions. This work utilizes direct numerical simulations to explore the mechanisms of growth of stationary streaks at the step and provides insight into their origin, nature, and spatial organization. The analysis is mainly focused on, but not restricted to, incoming perturbations in the form of stationary crossflow instability. Stationary streaky structures are found to be universal to swept forward-facing-step flow subjected to three-dimensional perturbations in the incoming boundary layer. The streaks at the step are primarily ascribed to the lift-up effect. They appear as a linear perturbation response of the highly sheared step flow to the cross-stream pattern of incoming perturbations. A mechanism of base-flow deceleration additionally contributes to feeding growth to the streaks. Linear stability analysis carried out through the harmonic Navier-Stokes method confirms that the streaks are a linear perturbation phenomenon. Effects of spanwise perturbation wavelength and effective sweep angle on the mechanisms of the streaks are also assessed.