Practical example of stabilizer codes Given the Steane code
$$
\left|0\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|0000000\right\rangle + \left|1010101\right\rangle + \left|0110011\right\rangle + \left|1100110\right\rangle + \left|0001111\right\rangle + \left|1011010\right\rangle + \left|0111100\right\rangle + \left|1101001\right\rangle)
$$
$$
\left|1\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|1111111\right\rangle + \left|0101010\right\rangle + \left|1001100\right\rangle + \left|0011001\right\rangle + \left|1110000\right\rangle + \left|0100101\right\rangle + \left|1000011\right\rangle + \left|0010110\right\rangle)
$$
and its relative stabilizers:
$$
K^1 = IIIXXXX
$$ 
$$
K^2 = XIXIXIX
$$
$$
K^3 = IXXIIXX
$$
$$
K^4 = IIIZZZZ
$$
$$
K^5 = ZIZIZIZ
$$
$$
K^6 = IZZIIZZ
$$
The stabilizer set establishes valid codewords for a state if the equation $$s\left|\psi\right\rangle=\left|\psi\right\rangle,\;\;\;\forall s \in S \;\;\;\;\; (1)$$
is satisfied. That means $\left|\psi\right\rangle$ is a +1 eigenstate of $s$.
We then consider a practical example of the usage of these stabilizers

The state of the system is represented by:
$$\left|\psi\right\rangle_F={1\over 2}(\left|\psi\right\rangle_I+U\left|\psi\right\rangle_I)\left|0\right\rangle + {1\over 2}(\left|\psi\right\rangle_I-U\left|\psi\right\rangle_I)\left|1\right\rangle$$
where $U \in \left\lbrace K^1,K^2,K^3\right\rbrace$.
We apply $U$ to the input state and we measure the ancilla qubits (syndrome measurement) to verify the integrity of the input (if $\left|\psi\right\rangle_I$ is +1 eigenstate of $K^1,K^2,K^3$). If the equation $(1)$ is not satisfied, then the corrupted qubit is corrected with a $Z$ gate addressed by the syndrome measurement. 
This is how does the system work?
 A: It is correct. We may summarize all the operations :
1) Encoding one logical qubit as $n$  physical qbits (codeword) , $\alpha|0\rangle + \beta |1\rangle \to \alpha|0\rangle_L + \beta |1\rangle_L$, here $n$ = 7 for the Steane code.
2) Preparing $m$ ancilla qbits, here $m = 3$, in your schema allowing to detect phase-flip errors $Z_i$
3) During transmission of the codeword, ther is exposition to noisy environment, and the codeword may suffer errors (bit -flip $X_i$, phase-flip $_iY$, bit-phase flip $Z_i$). In your schema, we are only intereseted in   phase-flip errors $Z_i$.
4) With the help of the generators, we compute the syndrome and store it in the ancilla qbits without altering the $n$-qbit world. 
5) Use the information provided by the error syndrome to locate the error  of any one of the $n$ qbits of the codeword.
In your case, the generators $K_1, K_2, K_3$ are the only needed to check any error $Z_i$, and each syndrome $xyz$ identifies precisely the bit $i$ corresponding to the error $Z_i$. Here the syndromes for the $Z_i$ are $010,001,011,100,110,101,111$
6) Correct the error 
Ref : Marinescu/Marinescu, Classical and Quantum information, Elsevier, p $462$, p $509$
