Shield platform dropping is an advanced technique in all four Super Smash Bros. games. It involves dropping through a soft platform while shielding by tilting the analog stick down.
In Super Smash Bros.
The technique is simple to execute in Smash 64, as there is no sidestepping in that game to interfere with the function of pressing the stick downward while shielding.
In Super Smash Bros. Melee
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As the sidestep is introduced in Melee, pressing the control stick too far and too fast results in a sidestep. Since this makes it hard to platform drop while shielding, the player must either be very precise with their inputs or press the stick down during a different action than shielding (such as a dash or shield stun) with close enough timing that the sidestep window is closed but the drop window is still open.
In Super Smash Bros. Brawl
In Brawl, while the conflict between sidestepping and dropping persists, there is now a small window where a drop can be made without triggering a sidestep: the player must move the control stick down to 68.75% of its maximum downwards angle in 6 frames without moving it past 70% of said angle in 4 frames.
In Super Smash Bros. 4
In Smash 4, the technique remains the same as in Brawl.
Uses
Shield platform dropping opens a larger possibility of comboing, tech-chasing, edgeguarding, mindgaming, and out-of-shield options. The player can run and pass under the platform, which cannot be done without shielding in Brawl. This technique also gives Yoshi a quick way to act out of his unique shield, as he cannot jump out of shield in Melee and Brawl, although it is less useful for him in Smash Bros 4 as he can jump out of shield once again.