There is some outwards flow, but most of the flow will be forwards. There is an increase in pressure from behind the fan to in front of the fan, but the blades move fast enough to capture the air and force it to flow forwards instead of backwards despite the pressure jump. Behind the fan, the air accelerates towards the fan as it's pressure decreases below ambient. As the flow passes through the fan blades, the pressure increases above ambient, but the speed of the air right at the fan blades remains about the same. Then the air continues to accelerate as it's pressure decreases from above ambient back to ambient.
As to why the air flows forwards instead of outwards, it's because of the angle of the blades. If the angle of the blades was perpendicular to the direction of rotation, the flow would be outwards and not forwards, but the angle is much less than this, so the direction of the "pull" and the "push" from the blades is mostly forwards and somewhat circular and/or outwards.