In general, basicity decreases with increasing atomic radius. However, this effect is apparent only when going up and down in the periodic table; moving across a row in the periodic table, the changes in atomic radius are not significant compared to other factors.
Now, why would atomic radius affect basicity? The answer has to do with charge density. Both the nitrogen in NR3 and the phosphorus in PR3 are slightly negative due to the presence of three alkyl groups (recall that alkyl groups are electron-donating groups). However, the effect of the electon donating groups is more pronounced in nitrogen. Why? Because of phosphorus's larger atomic radius, it's electron cloud has a larger area over which the negative charge can be dispersed. Nitrogen, on the other hand, has a smaller atomic radius and therefore a smaller volume. Thus, the negative charge donated by the alkyl groups is more concentrated in nitrogen's smaller electron cloud.
So, the charge density in NR3 is higher and there is a greater driving force to get rid of this excess charge. Since NR3 gets rid of excess negative charge by bonding with a hydrogen, NR3 will have a stronger driving force to become protonated; therefore, NR3 is a stronger base than PR3.