stand, counter for vehicles
臺 centers on a raised platform or stand, and by extension a counter for vehicles or machines that rest on such a platform.
臺 is the traditional form of 台. It combines 高 (tall) abbreviated with 至 (arrive) and 之 (go), suggesting a high place one reaches. The modern simplified form 台 is now standard in Japanese.
The top 吉 looks like a lucky ornament, the cover 冖 is a roof, and 至 below is arriving. Picture a lucky raised platform with a roof where vehicles arrive and stand.
For ダイ, imagine a die-cast model car displayed on a stand: die -> ダイ, and the stand is a counter for vehicles.