The onset of a word is its initial sound and the rime is the string of letters that follows, as in m-ilk, where m is the onset and ilk is the rime.
Teaching children to blend and segment the onsets and rimes of words builds their phonological awareness -- their understanding of sounds in a language. Understanding sounds in words is an important first step for children to learn how to read. This is because they need an awareness of the sounds that make up words before they are able to connect these sounds to letters.
Another similar activity that also builds phonological awareness are segmenting and blending the syllables of multisyllabic words.
For example, in shop, the onset is sh and the rime is op: sh-op. Shop has the same onset as ship, but it has a different rime: sh-ip.