⺮ is the compressed top form of 竹 and tells you the sieve is woven from bamboo - the traditional Chinese sifter is a flat bamboo basket. Same radical heads 笑, 答, 笔, 篇.
师 supplies the sound, with a tone shift shī → shāi. The traditional form 篩 has 師 below - simplification kept the same phonetic in its simplified shape. No semantic link; pure sound borrowing.
