As usual, I haven't made things very clear. I'll have another try.
Dew point is the temperature at which a parcel of air can no longer hold on to its moisture; if the temperature drops to that level, precipitation will result. Strictly speaking, it is the point at which the rate of condensation of water vapour outstrips that of evaporation. Colder air holds less water than warm air, and therefore, as temperature increases, the relative humidity decreases. Relative humidity, expressed as a percentage, is the ratio of the partial pressure of the water vapour in a given parcel of air set against the equilibrium pressure in that same parcel. Equilibrium pressure is basically the point at which the rate of condensation equals that of evaporation and the air cannot hold any more vapour.
Combining these definitions of relative humidity and equilibrium pressure, it can be seen that the relative humidity is comparing how much water vapour the air contains with how much it could contain. What all this means this means is that, as the relative humidity increases, the gap between the actual temperature and dew point narrows as the air is getting closer to its equilibrium pressure - it's close to maximum saturation, and even a small drop in temperature will cause precipitation.