Plain and simple, transitional pieces are clothes that you can easily wear between seasons. You wear it when you layer in the winter, but you also wear it early spring.
The simplest example would be a simple black cardigan. You wear it over a long sleeve tee in the fall, then in the winter you add a thick jacket and scarf over that. Then in the spring you ditch the jacket and scarf and swap the long sleeve tee for a short sleeve v-neck. Not every piece of clothing you own will transition between seasons, but some pieces, such as a plain black cardigan, transcend seasons.
Transitional pieces are important to incorporate in your closet, especially if you're wanting to consolidate and cut down on the bulk of your closet.