For fluorescent fixtures to work correctly, they need both tubes good and the ballast inside the light to be good. If any of those parts are failing, the light will be flickering & dim, or may not light at all.
So if the light is doing that, one of those parts is failing. Most often it's a tube that is wearing out, so the solution is to replace the tube with one you know is good (like one that is working in another light fixture). If that doesn't fix it, check the other tube (they often wear out at about the same time, since they are always on together).
If both tubes are good (they work in another fixture), then the problem is probably the ballast. In most of those 2-tube lights, it's a little thumb-sized can that plugs into a socket in the fixture. Often this is visible without dis-assembling the light -- it is sticking out under the tubes. Most either pull out, or turn a quarter- or half-turn and then pull out. There are usually markings on them telling the type & size needed, but the easiest is just to take the whole can into the store and tell them you need a replacement for this. Should cost less than $10.
Another possibility (but rare) is that the contacts where the tubes go are corroded, so not making a good electrical connection. Rubbing them with mild sandpaper or a kitchen scratching pad can fix that. But be sure to the light is turned off before doing that! For indoor lights, this is real unlikely, in my experience.