So I'm not super familiar with quantum computation, but I did do my undergrad research in QM (specifically, how chaotic behavior depends on the scale of nonlinear quantum systems) and I can take some informed guesses about what these words mean. It's actually super cool!

In a superconducting circuit

A circuit is a loop of something. Probably a solid material, like a metal or carbon, though it could be something more exotic. A superconducting circuit means the electrons in that material move without any resistance. This tells us the circuit is probably very cold--superconductors tend to break down at warm temperatures, like the ones in your house.

conduction electrons

Conductors have electrons in them. Some are "stuck" to atoms, others get to move around. Conduction electrons are the ones that move.

condense into a macroscopic quantum state

Macroscopic means "big", and for QM, "big" means, like, more than a handful of atoms or particles. At least as far as QM is concerned, everything--rocks, electrons, photons, people, etc., has a quantum state, but we use the phrase "quantum state" to mean a state that's, like, WEIRDLY QUANTUM. For instance, a pencil sitting on your desk is normal. A pencil that's like, half on your desk and half on mine is "quantum". Condensing means the electrons are going to change from doing normal individual electron things into acting like some sort of Big But Weirdly Quantum system, likely as a group. Like a crowd becoming a flash mob, they might do some sort of synchronized dance, only except the dance involves, say, every dancer doing two or three or ten dance moves at the same time.

such that currents and voltages behave quantum mechanically [2, 30].

Specifically, we're gonna be able to see quantum effects like superposition in Big Things like "current" and "voltage". The circuit might be in a combination of 3 volts and 5 volts at the same time. Also some of those voltages might be partly real and partly imaginary. Long story.

Our processor uses transmon

What the fuck is a transmon? I had to look this one up; it's a way of making these qubits less sensitive to voltage fluctuations.

qubits [6]

Qubits are quantum bits. A bit can be either 0 or 1. A qubit can be 0 or 1 or (and this is the quantum part) any state in between. Let's call the 0 state |0>, and the 1 state |1>. A qubit can be |1>, but it could also be (1/sqrt(2) |0>) + (1/sqrt(2) |1>). We call that a "cat" state, incidentally, because it's "half 1, half 0"--like Schroedinger's Cat, half alive and half dead. Again, the coefficients here are, in general, complex numbers, but we're gonna gloss over that.

which can be thought of as nonlinear

Nonlinear means they don't respond linearly to some input. Ever had someone do a series of small, mildly annoying things, and at some point you snapped and yelled at them? That's called "going nonlinear".

superconducting resonators

Oscillators are things that vibrate, like strings. Resonators have preferred frequencies to vibrate at. I don't exactly know what this means in this context, though. I'm guessing the circuit has some preferred frequencies it really likes to oscillate at.

at 5 to 7 GHz.

Voltages or currents or whatever are gonna go back and forth 5-7 billion times a second. That's about the same frequency as wifi signals, or microwaves.

The qubit is encoded

A qubit is an abstract thing on a whiteboard. There lots of ways we could actually make a thing that looks like a qubit. "Encoded", here, means "turned into an actual machine you can build in a lab".

as the two lowest quantum eigenstates of the resonant circuit.

An eigenstate, loosely speaking, is a state that has nothing in common with any other eigenstate. For instance, if we wanted to measure a particle's position on a line, we could take x=0 as one eigenstate, x=1 as another, x=2.5 as yet another, etc etc. An infinite number of eigenstates. Quantum systems can be in any (well, normalized) sum of eigenstates. My cat loves being inside and outside at the same time, so they're always trying to occupy 0.2|x=0> + 0.6|x=4> + 0.2|x=5>.

An operator is a thing you can do to a quantum state. Think of operators like functions on values, if you're a programmer, or like matrices that can be applied to state vectors, if you know linear algebra. For instance, I might have a measurement operator, which I use to look at my cat. There's also a special operator called the Hamiltonian, which (loosely) tells you what a state will look like after an infinitely small step in time.

Each operators has associated eigenstates, and those eigenstates have a magic property: if you apply that operator to one of its eigenstates, you get back the exact same state, times some complex number, which we call an eigenvalue. This means eigenstates for the Hamiltonian are, in a sense, stable in time. When we talk about the eigenstates of a system, we usually mean the eigenstates of the Hamiltonian. They could also be talking about measurement eigenstates--I'm not sure.

For the Hamiltonian, eigenvalues are, for Really Fucking Cool Reasons, energies. When we talk about "the two lowest quantum eigenstates", we mean the two states with the lowest energy. So maybe the circuit's eigenstates are, I dunno, 5 Ghz, 6 Ghz, 7 Ghz, etc. We'd take 5 and 6 as our |1> and |0> states.

Each transmon has two controls

A control is a thing we can use to change the transmon.

a microwave drive

Something like the microwave in your kitchen, but very small, and probably expensive.

to excite the qubit

This probably means changing the qubit from |0> to |1>. Microwaves carry energy, right? That's how they heat food. If they microwave the circuit at the right frequency, that microwave energy probably helps it jump from a lower frequency/energy to a higher one.

and a magnetic flux control

This feels like something specific to transmons. Flux has to do with the density of stuff moving through a surface. Magnetic flux probably has to do with how strong and close field lines are in some part of the transmon machinery.

to tune the frequency.

How fast the circuit wobbles depends on a magnetic field, I guess?

Each qubit is connected to a linear resonator

Huh. So we've got nonlinear resonators (the qubits) connected to linear resonators (some sort of measurement device?)

used to read out the qubit state

We need a way to actually look at the qubits, and I guess the linear resonator does that. I assume that the linear resonator is isolated from the qubit during computation, and once the computation is over, it gets connected somehow, and vibrates at the same frequency as the qubit. That process probably "spreads out" the quantum state of the system, pushing it REAL CLOSE to an actual eigenstate of the measurement system, which looks like a probabilistic measurement of the actual qubit state.

Like... my cat could be 3/4 inside and 1/4 outside, so long as the room is really dark. If I turn on the light, suddenly my cat is coupled to a MUCH BIGGER system--the room, and that "quantum" state gets diffused into that larger system, in what looks like a measurement like "cat definitely inside". I don't know a simple way to explain decoherence, haha, but if you like math, try Percival's "Quantum State Diffusion".

Hope this helps, and I also hope I got at least some of this right. Maybe someone with a better/more recent command of QM can step in here.