That said, the feature has always worked for me - actually it works too well that I have to think before I use it. Because after setting it once, it seems like I never hear from that person again. So it feels like an unfollow for me. Maybe the functionality depends on how much you interact with the person on Twitter? I don't know. I don't interact with people much on twitter, so when I choose that option, I don't see any likes from that person ever again, and it feels like I see their tweets A LOT less often than desired.
1. Agreement with the tweet, directed at the author only.
2. A form of "bookmarking" tweets for later. It's no accident that for most of Twitter's life this wasn't called a "like" but a "favorite" (with a star instead of a heart symbol).
As for "some of their followers will see it" and "I don't know [how it works]": That is precisely the problem. Your timeline isn't determined by your choices and the choices of the people you follow. The "algorithmic" timeline also adds a lot of randomness and unwanted choices that you can't control.
As far as I can tell, it has never had any effect whatsoever. I still see the components I dismiss, and I still see likes from the same people. Possibly less often... but it's hard to tell when it's every time I open the page.
At this point I click it mostly out of stubbornness and perverse curiosity, to see if the company is at all capable of changing this awful course.
But in reality, in twitter, like is a soft form of retweeting and users know it. So if someone you follow uses the "like / favorite" feature it also kind of means they want people to see it. For bookmarks, there is a different bookmark functionality. On twitter, like is a low weighted retweet that does not show up in the user's profile directly.