It's mostly her OP passive and her insane conditional mobility that make her incredibly strong.
Here's her passive in a nutshell:
Deal %Max HP True Damage (with a bonus AD scaling) + get a level-scaling heal + get a level-scaling % Movespeed bonus. 0.5s cooldown, 15s trigger window.
It sounds like a joke, right? But it's not. That's really her passive.
Next, her Q:
Short dash with auto-target bubble at the end. Prioritises triggering passive when possible. Barely costs anything (20/25/30/35/40 mana) and gets 60% cooldown refund if you hit anything with it.
Her W:
Stand still and become immune to absolutely everything in the game for 0.75s. After that, deal damage + slow for 1.5s. If you block hard CC, you will stun for 1.5s instead.
Her E:
Get fat attack speed, slap them once for fat slow, slap them twice for fat crit.
Her R:
Mark someone with 4 thingies. If you hit one + they die OR you hit all 4, you get a healing field stronger than Janna's ult + fat movespeed within the field.
New Fiora is OP because of her sheer versatility. She has severely overloaded 1v1 dueling options, she has lane sustain, she has fairly safe trading options in lane, she has teamfighting options... Really the only things she doesn't have are kiting options and reliable hard CC.
Obviously, there are counters (as Sky of Smiles pointed out), but they're mostly very niche picks that the vast majority of players aren't likely to have at hand.