I would certainly read it as:
Start accelerating at the accelerando so that you are playing at a moderato tempo when you get to the moderato, then continue to play moderato until you reach another tempo marking or the end of the piece.
That might not be exactly how others play it, and every performance involves interpretation of the score by the performer(s), but that is the only literal interpretation I can think of for those markings.
Moderato indicates a tempo faster than andante and slower than allegro, usually between 108 and 120 beats per minute.
Slowly could also be called lento and is likely meant to be 45 - 60 beats per minute. If so, that accelerando is really brief, since at the end of it you are being told to play twice as fast as at the beginning. Slowly might instead mean more of an adagio tempo of 66 - 76 beats per minute. Before you get too caught up in exactly how slow or fast to play, read the next paragraph.
Note the word freely in the initial tempo marking. That means you are invited to play around with the tempo a little bit. The composer expects/suggests that you speed up and slow down a little bit as you play to better convey emotion. A good question about that is, should we continue to play freely through the whole piece, even after other tempo markings? I would say yes, and I would interpret the freely at the beginning to indicate the whole piece is to be played with interpretive tempo.