The other one in the same opus is also a "Sonata quasi una Fantasia" and is more obviously one. It doesn't break down cleanly into movements but appears to start a movement, suddenly go into another one, and then come back to the first one again. A "Fantasia", for instance one of Mozart's, has a kind of "anything goes" improvisatory feel, where a "movement" doesn't really end but just before the final chord has a transition to a completely different idea. Often the first idea comes back later in the piece.
The "Moonlight" is less of a "Fantasia"; maybe Beethoven just wanted to have two pieces named "Sonata quasi una Fantasia" in the same opus number. There are two Fantasia-like things about the Moonlight: 1. The first movement really is an oddball piece. Sonata first movements are usually fast, dramatic pieces. This one is more-or-less in "Sonata Form" but not like any other sonata or symphony first movement. 2. Each movement is supposed to be played immediately after the previous one without pause (although there are no transitions as in a fantasia).
Now for performance practice: Look up what Andras Schiff has to say about this piece, if you can find it. He gave a series of lectures in London that were recorded and available online from The Guardian (newspaper in England) for a short time. Most pianists play the first movement much too slowly. Note that it is marked in 2/2 time (C with a line through it, sometimes called "cut time"). Adagio sostenuto, two beats per measure. The figuration (triplet accompaniment, "dunt - da - Dah" in the treble) comes originally from a short passage in Mozart's "Don Giovanni". (Schiff says that as a student Beethoven copied this passage by hand; the copy still exists in a museum.) The piece is actually a funeral march, not four steps per measure but a very slow two steps per measure. According to someone (Czerny? I forget who), Beethoven himself played it at half-note = 60.
Beethoven tells you to keep the damper pedal down for the entire first movement, and he means it. Should this be done on a modern piano? Well, the typical good-quality grand pianos of Beethoven's day had the same sustain time as modern pianos. I know because I've played some of them at the Frederick Collection. (Schumann's "Papillons" shows that his pianos had long sustain times too.) Beethoven uses the effect in another sonata as well. He liked this effect: it's like playing music or speaking in a vault or cave. Perhaps this is a funeral march that marches right into a large mauseleum. Remember that it also must be played very delicately and always pianissimo. He says so right there in the first measure.