If I pick an interval uniformly from a pool of intervals of which 95% contain the true value, it has a 95% chance of containing the true value.
Just like if I pick a ball from a bag of balls of which 50% are red, it has a 50% chance of being red.
Just like if I pick a ball from a bag of balls of which 50% are red, it has a 50% chance of being red.