drawer, tray, pad, screen
屜 centers on a flat, often removable surface used for holding or separating things: a drawer, tray, pad, or screen. The common idea is a thin, layered object that slides or rests in place.
屜 combines 尸 (a covering or shelter) with 彳 and 世, but the exact historical development is uncertain. It likely relates to objects placed under or within something, like a tray or drawer.
Under the shelter of 尸, the 彳 and 世 suggest a layered, flat object like a tray or screen that can be slid in and out.
For テイ, imagine a tea tray sliding out: tea -> テイ, and the tray is a flat pad for serving.