Twitter does have this functionality - it's called a ReTweet. Terminology differs, but it's essentially the same thing - both likes and retweets show up in the user's timelines, and both indicate liking what someone else has said.
Your followers probably follow you because your tweets are interesting to them. If you see something that you find interesting then chances are good that at least some of your followers will agree. Some followers may have seen it from the original source; others may only get it from your retweet.