There are many contributing factors:

• Longer focal-lengths require faster shutter-speeds to reproduce details sharply when hand-held. The general rule-of-thumb is 1 over the effective focal-length of the lens. So a 300mm on a DX camera has an angle of view equivalent to 450mm and so you should expect 1/500s at least to get sharp images. The solution against this problem is simple: Use a tripod or a faster shutter-speed, exposure permitting.
• Should you have the Nikkor AF-S 70-300mm F/4.5-5.6D ED, the optics are really poor towards the long end. Lenses do not have exactly the same performance at all focal-length. In particular that lens is extremely soft at 300mm at F/5.6-8. Things get better at F/11 but never quite match the performance at 70mm F/8 where the lens performs optimally. Should you want to visualize what is happening, check this out.
• This is a variable aperture zoom, so at 300mm you can only open the lens to F/5.6 which is not even desirable due to optical performance but already reduces the amount of light reaching the sensor which causes shutter-speed to lower. This can be partially compensated by increasing ISO but that introduces noise which destroys fine details after a point.