First of all, the phrase „I am in a spaceship travelling at close to the speed of light“ strictly speaking does not make sense. Why not? Because you cannot simply travel at close to the speed of light. You can only travel at close to the speed of light relative to something. In your case, the spaceship travels at close to the speed of light relative to the earth. On the other hand, the earth “travels” at close to the speed of light relative to your spaceship. The situation is symmetrical. This is crucial for understanding relativity.
Secondly, the phrase “time within my spaceship moves slower than that of another person on earth” does not make sense either. Why not? Because you are comparing apples and oranges. In order to compare the time passing on your watch with the time a person on earth sees on his watch passing you need to define simultaneous events you both agree upon that they happened at the same time (you need to agree upon the point in time you both need to take a look at your watches in order to compare your readings). This however, is not possible: According to relativity, you will not find agreement: Events which the person on earth sees as happening at the same time you will not consider as happening at the same time (provided that the events occur at different locations).
What you probably mean instead is the following: The time in your spaceship moves faster than the time on a clock located on earth as you observe it. Please note the difference: In the first case the person on earth checks the time on the clock on earth, in the second case you are watching the clock on earth from your spaceship. This makes a crucial difference.
Of course, when you observe the clock from your spaceship, you will receive the time from this clock only at a delay since moving at close to the speed of light your distance to the clock will increase (or decrease) rapidly and since nothing can travel faster than light the info about the time on this clock will not arrive at your spaceship faster than that. After you correct the incoming signals for this delay you will find that the the clock on earth moves slower than your clock.
On the other hand – the situation is symmetrical – the person on the earth will find out the same thing about your clock in the spaceship as he observes it: Also, this clock is moving slower relative to his clock.
How does this fit together? Isn’t this a contradiction? In one case, your clock is going slower, in the other the clock on earth? Again, you are comparing apples and oranges: Since you and the person on earth will never agree on the simultaneity of events when they happen at a distance, you cannot compare the readings of you and the person on earth directly.
But of course, you can return your spaceship, go back to earth and compare the clocks directly:
Guess you started from earth, made sure that your clock and the clock of the person on earth were synchronized. After that you travelled to a distant start and now you return, always travelling at close to the speed of light relative to the person on earth. After your return you will find out that on the clock on earth much more time has passed than on your clock. For the person on earth this sounds reasonable: He saw your clock running slower all the time during the trip. But you also saw his clock running slower all the time during the trip. Where did the extra time go that passed on the clock on earth? Where did you miss something?
As I said in order to read and compare the time you need to have a concept of simultaneity. You can define a consistent concept of simultaneity within your spaceship frame while you are travelling at constant speed: So you can compare your local time with the time you see on the clocks on earth. Likewise the person on earth can define for himself in his frame which is at rest relative to the earth a consistent concept of simultaneity. This concept, however, will not fit to your concept. The more far away from each other events occur the more you and the person on earth will disagree about the time difference between these events.
Now, at some point on your journey you need to return: you need to decelerate your spaceship and accelerate it in the opposite direction in order to be able to return to earth. During this period of deceleration and acceleration your concept of simultaneity “changes”. This involves events which happen far away from you (e.g. on earth). That means - during your deceleration period - an event on earth you regarded as simultaneous to your here and now you will not anymore regard as simultaneous a second later but as happening some time, maybe hours, days or even years ago depending on the distance and the strength of your deceleration. That, essentially, is were the “lost” time is going. And this is why the twin paradox at the end can be resolved.
