There are many examples of the physical meaning of an inflection point.
If $y$ is the position of an object in one-dimensional motion, the inflection point is where the force on the object changes direction.
Here are some economic applications. If $y$ is the total cost of producing $x$ number of units of a product, the inflection point is where the marginal cost reaches a minimum (or perhaps maximum).
The growth of a company often follows a logistic curve. If $y$ measures the size of a company in any sense, the inflection point is where the growth is at a maximum. Similarly, the inflection point shows the maximum spread of a sickness, which also usually follows a logistic curve.
Here is a real-life example I show in class. I cut the following graph at $x=22$ and ask the class what is happening to the employment. The inflection point of the "current unemployment recession" is at about $x=12$, so that is where the rate switches from getting worse and picking up speed to getting worse but we're about to turn the corner. That point was into Obama's presidency, so after that point Obama was claiming that he was helping the economy even though employment was still getting worse. The politics is debatable but that does follow the graph.
I summarize with a quote from Hugo Rossi. "In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection."