It came to prominence as an LCS counterpick to Zyra and 2-0'd her. When Malzahar became a meta support, she was also good against him (although Zyra is usually still considered a better counter to him). She's mostly a counterpick but can punish other match-ups as well.
The positives:
• She has an incredibly easy and reliable gold-generation pattern for a support in the form of Make it Rain + Spellthief's Edge, so she'll usually be ahead in income unless something goes really wrong in the lane.
• Near-instant Thunderlord's Decree proc off Make it Rain. It's even faster than Morgana's Tormented Soil — damage every quarter second vs. every half second.
• Once she has Black Cleaver, Bullet Time starts demolishing people. Combine it with any snare or a strong slow in a team-fight, non-tanks will either die or have to burn a lot to get out.
• Make it Rain grants vision, so you can scout with it.
• Mana regen off the Spellthief's Edge line and Meditation will largely mitigate the mana issues MF is known for.
• Can soften up a wave very easily with Make it Rain, lets you push quickly.
• She has more pressure on turrets than other supports because she usually stacks AD. Besides that, she has a built-in attack speed steroid, for even more turret pressure. With bot lane turret lacking the early-game bonus resistances, this is really strong.
• If need be, she can get Rylai's and Liandry's on top of Black Cleaver to not only shred Armor, but melt people with Make it Rain and Bullet Time. Requires a lot of gold, but again, you get it fairly easily.
• Naturally fast thanks to Strut; very useful for getting back to lane, dipping out of lane to ward, etc.
• Punishes players that try to hide behind minions with Double Up.
• Lethality is a thing. (This is why she's come back into play as a regular bot laner, too.)