When you're comparing bond angles between $\mathrm{NH_{3}}$ and $\mathrm{NF_{3}}$, you'd want to take the electronegativities the of hydrogen and fluorine into consideration.
Fluorine hits a 3.98 on the Pauling Scale for electronegativity, while hydrogen does a 2.2 on the same scale.
Hydrogen, with its said electronegativity of 2.2 does a fairly decent job of pulling the N-H electron bond pairs towards it. As you would know, adjacent electron bond pairs repel, and it's this repulsion that gives ammonia an H-N-H bond angle of 107°.
Now fluorine, with an electronegativity of 3.98, does an even better job of pulling the N-F electron bond pairs towards it. Since its got a much greater electronegativity than hydrogen, fluorine's able to pull out the electron pairs even further. The consequence of this is that the N-F bonds can be pushed closer than N-H bonds before they're stopped by the repulsion of their electron bond pairs. This is what gives $\mathrm{NF_{3}}$ a smaller bond angle of 102°.
