I good way to understand this circuit is to try and implement it in Stim.

1. Why does measuring the $\vert +\rangle_L$ state in $Z$ basis (left half of figure) give us the correction $X_i$?

$X$ errors anticommute with $Z$ measurements, so from the measurement results you can figure out the correction.

2. How do I use the ancilla measurments of the figure for decoding? The figure suggests that I can simply correct any individual $X_j$ or $Z_i$ physical error on the data qubits after measuring the ancilla.

You have to combine measurement outcomes to form detectors. I wrote an introduction to detectors in Section 2 of this paper. In this circuit, the 7 Z basis measurements of the ancilla qubits can be used to form 3 weight 4 detectors. Same goes for the 7 X basis measurements. You can, although this is not precise, think of these detectors as the measurement results of the stabilizer generators of the Steane code.

3. Do I even measure the stabilizers of the code used to encode $\vert\psi\rangle_L$ at any point? Or is that information contained in the ancilla measurements?

Measuring ancilla qubits individually and combining the measurement results is equivalent to measuring stabilizers directly.

A good way to understand this circuit is to try and implement it in Stim.

1. Why does measuring the $\vert +\rangle_L$ state in $Z$ basis (left half of figure) give us the correction $X_i$?

$X$ errors anticommute with $Z$ measurements, so from the measurement results you can figure out the correction.

2. How do I use the ancilla measurments of the figure for decoding? The figure suggests that I can simply correct any individual $X_j$ or $Z_i$ physical error on the data qubits after measuring the ancilla.

You have to combine measurement outcomes to form detectors. I wrote an introduction to detectors in Section 2 of this paper. In this circuit, the 7 Z basis measurements of the ancilla qubits can be used to form 3 weight 4 detectors. Same goes for the 7 X basis measurements. You can, although this is not precise, think of these detectors as the measurement results of the stabilizer generators of the Steane code.

3. Do I even measure the stabilizers of the code used to encode $\vert\psi\rangle_L$ at any point? Or is that information contained in the ancilla measurements?

Measuring ancilla qubits individually and combining the measurement results is equivalent to measuring stabilizers directly.

I good way to understand this circuit is to try and implement it in Stim.

1. Why does measuring the $\vert +\rangle_L$ state in $Z$ basis (left half of figure) give us the correction $X_i$?

$X$ errors anticommute with $Z$ measurements, so from the measurement results you can figure out the correction.

2. How do I use the ancilla measurments of the figure for decoding? The figure suggests that I can simply correct any individual $X_j$ or $Z_i$ physical error on the data qubits after measuring the ancilla.

You have to combine measurement outcomes to form detectors. I wrote an introduction to detectors in Section 2 of this paper. In this circuit, the 7 Z basis measurements of the ancilla qubits can be used to form 3 weight 4 detectors. Same goes for the 7 X basis measurements.

3. Do I even measure the stabilizers of the code used to encode $\vert\psi\rangle_L$ at any point? Or is that information contained in the ancilla measurements?

Measuring ancilla qubits individually and combining the measurement results is equivalent to measuring stabilizers directly.

I good way to understand this circuit is to try and implement it in Stim.

1. Why does measuring the $\vert +\rangle_L$ state in $Z$ basis (left half of figure) give us the correction $X_i$?

$X$ errors anticommute with $Z$ measurements, so from the measurement results you can figure out the correction.

2. How do I use the ancilla measurments of the figure for decoding? The figure suggests that I can simply correct any individual $X_j$ or $Z_i$ physical error on the data qubits after measuring the ancilla.

You have to combine measurement outcomes to form detectors. I wrote an introduction to detectors in Section 2 of this paper. In this circuit, the 7 Z basis measurements of the ancilla qubits can be used to form 3 weight 4 detectors. Same goes for the 7 X basis measurements. You can, although this is not precise, think of these detectors as the measurement results of the stabilizer generators of the Steane code.

3. Do I even measure the stabilizers of the code used to encode $\vert\psi\rangle_L$ at any point? Or is that information contained in the ancilla measurements?

Measuring ancilla qubits individually and combining the measurement results is equivalent to measuring stabilizers directly.