How do I make sense of the clock example without getting lost in the math?

The key distinction is between what someone sees through incoming light and what they calculate happened at a particular time. Light takes time to travel, so a distant observer does not watch the whole trip unfold in real time. After correcting for that delay, observers in different frames can still assign different elapsed times because they disagree about which distant events are simultaneous.

For the simple version, take the stars to be 10 light-years apart in their rest frame and the ship's speed to be 0.9 times light speed. In that frame, the trip takes about 11.1 years. The crew's elapsed time is shorter, about 4.8 years. Those are different measurements of time along different paths, not a claim that anyone's clock locally runs strangely.

One part of the setup is misleading: a clock does not tick because it travels at the speed of light. Light has the same measured speed for inertial observers, but ordinary clocks measure elapsed time. A light-clock thought experiment can help explain the result, though it is a model of a clock, not a rule that every clock is a beam of light.

Also, the crew's shorter elapsed time is not explained by light being slower for them. They still measure light at the same speed. The differences in measured distance and time between frames fit together so that this remains true.

Intuition is useful for spotting what you think should happen, but it can quietly assume that all observers share one universal “now.” That assumption works well in everyday life and fails for distant events in relativity.

I would keep the story and the calculation side by side. First ask what each observer measures locally, then ask how they define the start and finish events, and only then compare elapsed times. If an analogy starts making light sound like a clock hand or makes a visual image equivalent to a measurement, it has probably stopped helping.