My answer comes years late but differs from the others.
A bishop can often reach a square from which it can control, restrict or bind an opposing knight. See the diagram, which gives an unspectacular but typical example.
[title "S. Mamedyarov v. M. Carlsen, Wijk an Zee, 2018"]
1.d4 Nf6 2.c4 e6 3.Nc3 Bb4 4.Qc2 d5 5.cxd5 Qxd5 6.e3 c5 7.Bd2 Bxc3 8.Bxc3 cxd4 9.Bxd4 Nc6 10.Bc3 O-O 11.Nf3 Rd8 12.Be2 Qe4 13.Rc1 Qxc2 14.Rxc2 Nd5 15.Ne5 Bd7 16.Nxc6 Bxc6 17.Bd2 Ne7 18.f3 Rac8 19.e4 f6 20.Be3 a6 21.Kf2 Bb5 22.Rhc1 Rxc2 23.Rxc2 Bxe2 24.Kxe2 Nc6 25.b4 Rc8 26.Rc5 b5 27.f4 Kf7 28.a3 Ne7 29.Rxc8 Nxc8 30.Bc5 Ne7 31.Bxe7 Kxe7 32.Kf3 Kd6 33.Ke3 e5 34.f5 Ke7 35.g4 Kf7 36.h4 Kg8 37.Kf3 h6 38.h5 1/2-1/2
White moves 30.Bc5. In this particular game, Black is Magnus Carlsen, so Black has foreseen the trouble and has advanced his king into a position from which the king can help to rescue the knight from its bind. The point, however, is that the knight is indeed in a bind. The bishop has done this to the knight.
Admittedly, knights can take away squares from bishops, too, but a bishop can often escape by advancing past the knight out of the knight's range. On average, bishops have the upper hand over knights in positions like these.
Nor are positions like these rare. Such positions are common, in fact. Indeed, such a position occurs again earlier in the same game. (Easy exercise: find the position. The same white bishop restricts Black's other knight for six full moves.)
This bishop-versus-knight dynamic, in which the bishop has slightly the better of the struggle, is a significant reason to regard the bishop as being—on average—slightly stronger than the knight.
The knight can of course reach all squares of the board. This is a significant point in the knight's favor, but the bishop has other, subtle advantages. For instance, a bishop can three-step.
Consider: a knight can move away from its square and back in two moves but, interestingly, cannot do so in three moves. A bishop by contrast can indeed use three moves to return to the square from which it starts. In an endgame, this three-step dynamic can allow the player with a bishop to put his opponent, who has a knight, in Zugzwang.
There is also the bishop pair. One of the other answerers opines that nonexpert players can seldom effectively use the bishop pair. Actually, there is some truth in that opinion, but I have (occasionally) seen nonexpert games decided by a pair of bishops, so you have that factor to consider as well.
On the edge of the board, bishops are often happier than knights.
A bishop and a pawn can mutually protect one another—yet, with a quick push of the pawn, the position can be converted into one in which the pawn covers squares of one color and the bishop squares of the other.
All factors considered, on average, the bishop is probably very slightly the better piece. But yes, of course, as others have pointed out, it depends on the specific position at hand.