No, it does not mean that time speeds up when you're in motion. The phenomenon you're referring to is called time dilation, which is a fundamental concept in special relativity. Time dilation states that time appears to pass more slowly for an object or observer that is in motion relative to another object or observer.
In the case of the photon clock scenario, let's say there is a photon bouncing between two mirrors inside the clock. For a stationary observer, the speed of light is constant, and the time it takes for the photon to complete each round trip is always the same. However, when the photon clock is in motion relative to an observer, the observer will perceive the path of the photon as longer due to the effects of relative motion.
According to special relativity, the speed of light is constant for all observers, regardless of their relative motion. To reconcile this constancy of the speed of light, time itself must be affected. As the observer sees the photon travel a longer distance, they also see it take longer to complete each round trip. In other words, time appears to slow down for the moving clock from the perspective of the stationary observer.
From the point of view of the passenger on the moving train, their experience of time remains normal, and it is the stationary observer who perceives time as ticking slower for the moving clock. This phenomenon has been experimentally verified and is a fundamental aspect of our understanding of space and time in modern physics.