Why is the channel length modulation effect more visible in short-channel devices?

When we reduce the channel length, the drain depletion region eats into the channel more aggressively. This reduces the control of the gate and makes the current increase even after saturation, which is exactly channel length modulation.

we should also look at it practically. In long-channel devices, the channel length is large, so a small shrink at the drain side does not change much. But in short-channel devices, that small shrink is a big percentage of the channel, so you see the effect clearly in output characteristics.

Channel length modulation is more visible in short-channel devices because the effective channel length changes a lot when the drain voltage increases. Since the channel is already very short, even a small change in depletion region at the drain side has a big effect on current.