Where does time goes when I travel through space?

I asked my first question on “quora” and observed even, how bad my English is. Hack! Now, that is my question: What happens to the “old” images of a distant planet in space when I fly there and arrive and see “new” images? How do they become gradually “younger”?

What happens to the images of a far away point in the universe, if I am travelling towards it? From earth I can see that point, lets call it “planet X” in my telescope and I know those images are actually “old”. Happily flying towards planet X, arriving after some time, there is no time-shift left, the images have become “contemporary”.

Now: What happened that I now see that planet X in “real time”? – Whereas before it were “old images” … and while travelling those images of the planet X became continously younger and younger?

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