People can see both direct and reflected light. Perhaps whoever told you that we can only see reflected light meant that most of the objects around you are visible because they're reflecting light from some other source. For example, the light emitted by the fluorescent tubes in the ceiling of my office is bouncing off my coffee mug and into my eyes, allowing me to see the mug. The fluorescent tubes, on the other hand, are emitting their own light. You can see that light directly, although it might scatter off the glass that encloses the tube, the gas molecules in the air, or off the surfaces inside your eye, and be "reflected" in that way.
So there are basically two categories of things you can see: things that reflect visible light (coffee mugs, people, the earth) and things that produce visible light (fluorescent tubes, light bulbs, a flame, the sun, computer screens). Either way, some of the light has to actually enter your eye. You can see a light bulb from any position because it emits light in many directions. The same goes for a coffee mug illuminated by a ceiling light--it's basically reflecting light in all directions.
A laser beam, on the other hand, is concentrated in one direction. It's invisible unless it is pointed directly at your eye (don't do that!) or if some of it bounces off something like dust particles in the air, reflecting a little bit of light towards you. If a beam of light is just traveling through space without reflecting off anything or hitting your eye, you won't be able to see it.