When a Svelte component with a transition mounts, it first checks the delay parameter. If delay is set, Svelte waits that amount of time before starting the animation. After the delay, the animation runs over the duration time, gradually changing the component's appearance, such as fading in. For example, with a delay of 500 milliseconds and duration of 1000 milliseconds, the component stays hidden for 500ms, then fades in over 1 second. Variables like delayTimer track the waiting time, animationProgress tracks how far the animation has run, and divOpacity changes from 0 to 1 during the fade. This process ensures smooth, timed transitions that can be controlled precisely.