There are actually several champions that make it work such as Ezreal (active procs Muramana), Twitch (one of the main BotRK users), Lucian and Kalista (Likes to get down with AS). Probably not THE best pick up on Lucian since Critty Edge is more in line with his bursty habits, but it still provides Lucian comparable DPS with Bloodthirster.
After the pretty significant buff of having life-steal apply on the % damage and cast range being buffed up to be the same as the average ADC range, it increases your damage output and sustain significantly against high-HP targets. If you're using Ezreal for example, it makes a great poke down tool against the frontline champions setting them up for an easy kill by your teams mage or assassin while with Twitch it makes a great backline assassination tool and punishes champions for pushing lanes alone.
So really, it comes down to a situational pick based on champion, role and what you're aiming for. With Vayne, it gives her dueling potential. Ezreal gains better poke, Twitch gains better assassination (and subsequently better teamfighting presence when used with Rat-a-ta-ta), Lucian levels his Damage Over Time, and Kalista just likes launching spears faster.
What really puts BotRK down on the tier list is the current popular ADCs which is Jinx, who doesn't need it for obvious reasons, Kalista, who enjoys other items just as much or more, Graves who relies more on burst, Sivir who doesn't need the additional AS (them ricochet resets), and Corks, who can't proc it with missiles (another reason why he isn't a candidate for Muramana.)
TL:DR
Depending on the champion, items yield different results.