Please make sure you understand that it is a confidence interval and not a probability as the true value of the population parameter either lies in the interval or not.
From the diagrams below, you will see the theory of how an interval is created by using the area underneath the curve and the symmetry of the curve.
The standardised curve is used so the centre value is Zero rather than the value of the Population Mean.
So these Z -values will be change depending on the area trapped but will always be the same value regardless of the value or nature of the Population parameter.
For example, if we wanted a 95% confidence interval of a population proportion then
Z-value would be 1.96 provided we had a normal distribution.
You will see that not all the intervals created will contain the true value of the population parameter (this example is based on the Population Mean though any measure may be used).