This is a labelled toy model, not a spaceship. One honest idea runs everything: a clock made of bouncing light ticks slower when it moves. The flash always travels at the same speed - that is the one strange, extremely well-tested rule - so when the mirrors slide sideways and the bounce becomes a longer zigzag, each tick must take longer. The arithmetic here is exact special relativity (the stretching factor is the real gamma), and you can audit every tick.
What is real: super-accurate clocks flown around the world on jets really come back slightly behind clocks that stayed home, exactly as the rule predicts; GPS satellites fly so fast that their clocks must be corrected continually or positions would drift; and the speed of light really is the same for every watcher - that part is an experimental fact this toy asserts rather than proves.
Left out on purpose: the toy cart rides at 80% of light speed so the slowing can be SEEN (real-life effects are billionths of a second - the scene says so), the mathematics of gamma is not taught, gravity's separate slowing of clocks belongs to a later lab, and the subtleties of who-sees-what-when are saved for grown -up physics. No real journeys or personal details are used; nothing about you leaves this page.
Source: NASA Space Place ("Is Time Travel Possible?") - clocks on airplanes and satellites really run at a different speed than those on Earth, and GPS relies on these corrections every day. The link is in the project readme.