What exactly is it and how do I kill it

That's river cane or possibly switch cane. I'm not sure if switch cane grows here. I wish I lived nearby. I'd love to fill up a truck bed of that stuff. River cane makes decent arrow shafts and great atlatl darts. I don't know how to kill it except digging it all out. It spreads underground.

I wouldn't mind getting some starts of that and planting them at work.

Here you go it is phragmite. It can be killed but it will be back if you don't kill it all which is impossible.
en.wikipedia.org/wiki/Phragmites.
Peabody Coal has spent God only knows how much money trying to kill it off of their mitigation ditches they are required to put in.

It's also called common reed. It's pretty invasive and spreads by seeds or by stolons (roots growing and spreading).

It likes wet/damp areas and can grow so dense that critters can't walk thru it. It may take 2 applications, but imazapyr works VERY well in controlling it. Also whatever else you spray it on..... I would mix 1 fl. ox. of a non-ionic surfactant per gallon of tank mix, and follow the application rate in applying the chemical. Apply it when the plant is actively growing, so late Spring thru late Summer. Be sitting down when you get the chemical price..... The trade name of the chemical is Habitat. Here is some info on it:

Habitat contains the active ingredient, imazapyr, which inhibits the plant enzyme AHAS (acetohydroxyaced synthase). Habitat is a systemic herbicide that is effective on post-emergent floating and emergent aquatic vegetation. Imazapyr is effective at low-volume rates and does not contain heavy metals, organochlorides or phosphates, making it safe to humans and livestock. Habitat requires the use of a spray adjuvant when applying on post-emergent vegetation.

Here's a link to the label:

Habitat

Since the plant grows in moist/wet areas, you have to use a chemical that is labeled to be OK to use around water.

River Cane (Arundinaria gigantea), Switch Cane (Arundinaria tecta) and Hill Cane (Arundinaria appalachiana) all belong to the Poaceae, or Grass family of plants. Although they are referred to as “cane”, they are true bamboos closely related to those found in Asia and parts of South America. While there has been some confusion as to whether or not River Cane and Switch Cane are one and the same species, recent work has proven without a doubt that the genus Arundinaria includes three separate and distinct species, which together are the only temperate native species of bamboo found in the Northern Hemisphere.

If you want to get rid of it completely wouldn't you want to chisel plow it if you can, but it and then treat the soil that is turned over? I agree spraying it would be simpler, but I may be all wrong here but to completely get rid of it don't you want to "kill the infection" and "not treat the symptom". Curious to know how your eradication efforts go. Keep us updated.

Don't mix it too hot, and make sure to use a surfactant. If you mix it too hot the plant will shut down and not translocate the gly to the roots where it needs to go to kill it, all you will do is give it a haircut.