It’s a figure of speech. It’s a slight exaggeration for emphasis.
Let’s look at this example:
They had no sooner eaten dinner than the ceiling crashed onto the dining table.
That literally means:
They finished eating dinner no earlier than the ceiling crashed onto the table.
In other words, there was no time delay between the end of the dinner and the crash of the ceiling.
Really, there probably was some slight delay, or they wouldn’t have gotten away from the table in time. Someone might even say “no sooner than” if there was a delay of five or ten minutes. The exaggerated phrasing just means “Whew! That sure was close!”
You could argue that the idiom is also an understatement, because if they finished eating dinner “no sooner than” the ceiling crashed, the ceiling could have crashed a month, a year, or a century before they even started eating dinner, allowing plenty of time for repairs. People don’t hear it that way, though. I think this explains the peculiar pair of tenses in the idiom. The perfect tense emphasizes the time interval that contains eating dinner, and since it’s past perfect, the end of that time interval is considered in relation to “the ceiling crashed”, which is expressed in the simple past tense. “No sooner than” connects them, saying that the time interval for eating dinner ended exactly when the ceiling crashed.
Another example of the same figure of speech is:
No sooner said than done.
This literally means, “There is no time delay between your giving the order and the order being accomplished.” Of course that can’t literally be true, since no one can carry out orders instantaneously. The figurative meaning is just that there will be a very short time between giving the order and accomplishing it.