A confidence interval does not always "work" — it may not actually include the unknown parameter value. A 95% confidence interval might be
\[ 0.7773 \;\;\lt\;\; \theta \;\;\lt\;\; 0.9483 \]but the value of \(\theta\) is unknown and may not actually lie within this interval. The best we can say is that we are 95% confident that \(\theta\) will be between these two values.
If confidence intervals were found from other similar random samples, 95% of them would include \(\theta\).
The notion of a confidence level therefore more a characteristic of the method of finding the confidence interval, than a characteristic of the specific confidence interval from a single data set.
Simulation
The simulation below took 100 random samples of size n = 200 from a population with π = 0.6. Most of the confidence intervals included π = 0.6, but some did not. If the simulation was repeated many more times, the proportion including 0.6 would be close to 0.95.
In practice, you only have a single sample and a single confidence interval, but we have "95% confidence" that it will include the true (and usually unknown) value of π.