A "kill lane" in bottom is a different strategy of winning your lane. A very general definition of "winning your lane" is that you have generated more gold, experience, and map presence than your lane opponent(s).
In bottom lane, the standard strategy is to get a lane with high sustain and ranged attacks which reduces the potential for being harassed and reduces the impact of harassment by having heals, shields, or other damage mitigation. This is essentially a "farm lane." Your goal is to get as much creep score (CS) as possible, and hopefully some kills along the way to bolster the amount of gold and experience you receive (as well as reduce the amount your opponents receive).
By contrast, a "kill lane" is one where the focus is on killing your opponents. Usually this is accomplished by having two characters with crowd-control, at least one of whom can also deal fair damage. You would normally pick a kill lane to counter a farm lane, which often has at least one character considered "squishy" (low hp, armor, and/or magic resist), making the kills easier. Commonly chosen combinations are Jarvan and Leona, Sion and Nautilus, and (more antiquated due to Taric's lack of damage) Taric and Sion. The reasoning is that each of these groups either have a large amount of CC, burst potential, tankiness (so they can tower-dive and are hard to get out of lane), or a combination of all three.
Sion makes a great choice because of his stun and burst potential. You can also build AD Sion and rely on your extra damage from Enrage in combination with your stun. Leona has fair damage when paired with someone else because of her passive ability, and also has three crowd-control mechanisms at level 6. Jarvan is a good choice with her because he has two crowd-control measures of his own and has similar mobility. Nautilus is prized for the same reasons as Jarvan, but his passive gives him an additional crowd control opportunity (though he lacks the same damage and farming potential as Jarvan).
While it's obvious that you get gold and experience from repeatedly killing your lane, those numbers become less significant as the game goes on (assuming you're even successful). Additionally, at first glance, it seems like you wouldn't be able to generate as much gold for your lane as a strict farm lane could (given the diminishing returns of killing the same player over and over). However, there are important indirect results of the kill lane:
- First and foremost is that your opponents become easy to zone out of experience and CS even if you don't kill them. Killing them of course means they spend more time out of lane. So while you may not be generating as much if you were simply farming, they are generating far less as well.
- Killing your lane gives you the chance to gank mid, top, or counter-jungle which not only gives you some extra gold, but also increases the gap between your allies and their respective lanes.
- Repeatedly killing your opponents gives you more opportunity to push the tower which can be an important gold, experience, and map presence boost early on. It also aids in accessing the lower portion of your opponent's jungle.
- A kill lane plus your jungler should easily be able to take down dragon very early, again providing an important gold bonus to your team and helping to establish control of that objective.