Special relativity gives every mass in the universe its own clock. I have one, you have one, and so does every planet, but they only keep the same time if they have the same speed. General relativity adds that gravity alters time as well, so it takes a lot of computing to make GPS navigation work because the clocks of satellites tick at a different rates depending on their altitude and speed.
How then does the earth change the time of objects around it? Virtual time ticks by as processing cycles complete, so it runs slower as processing increases. The earth’s gravity increases the network load closer to it, so it slows down our time. A clock on top of the Empire State building then ticks faster than one at the bottom because the network load is less there.
Would we live longer on a planet like Jupiter that has more gravity? To others it might seem so but to us, time would pass as usual. A large planet dilates time relative to earth, but the number of life events in our lifetime wouldn’t change, so we wouldn’t notice it.
The earth’s gravity gives the space closer to it more to do so time passes more slowly there. We feel gravity as a force, see space as an extent, and experience time as life events, so why Einstein’s equations connect them isn’t obvious. In physical terms, gravity, space, and time correlate but in quantum terms, the same activity generates them all as effects not causes. Matter then doesn’t change space or time to cause gravity, but changes the quantum field that generates them.