Space itself is stretching, and it stretches the light waves travelling through it. A galaxy that is further away has more space between it and us, so its light is stretched more and has a bigger red-shift.
The faster a galaxy is moving away, the bigger its red-shift. Astronomers measure bigger red-shifts for more distant galaxies, so more distant galaxies are moving away faster. This is evidence that the universe is expanding, and that it began with the Big Bang.
This higher tier covers Higher Tier: Why Distant Galaxies Show More Red-Shift within Red Shift & Big Bang for GCSE Physics. Revise Red Shift & Big Bang in Space Physics for GCSE Physics with 13 exam-style questions and 12 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 11 of 14 in this topic. This section is most useful once the core foundation idea is secure, because it adds the detail that pushes answers higher.
Practice questions for Red Shift & Big Bang
What is the orbital period of a geostationary satellite?
Explain why a geostationary satellite stays above the same point on Earth's surface.