Why is Albanian so different from other European languages?

To expand on Edmond Pano’s answer:

Indo-European languages are not all that similar to each other. That’s why it took so long to establish the family. (It was much more obvious in Classical times, but people in Classical times weren’t paying attention.) The level at which laypeople can tell similarities is at the branch level.

So Danish isn’t from outer space if you’re aware of German, and Spanish isn’t from outer space if you’re aware of Italian, and Czech isn’t from outer space if you’re aware of Bulgarian. (Notice I didn’t mention French and English, which are still quite odd.)

But an isolate branch like Albanian or Armenian is going to stand out, because there’s no immediately close language. In fact the only reason why we don’t say that about Greek more is that people at large are already a little familiar with Greek, because of its cultural influence.

If you’re Greek or Macedonian, however, and leaf through an Albanian grammar, it doesn’t look different at all: the Balkan Sprachbund has made its grammar very close to its neighbours. And if you work out the sound changes, it’s surprising how much of the Albanian vocabulary is chewed-up Latin.