Increase in bond length of BF₄⁻ compared to BF₃
There seems to be a bit of confusion between bond length and bond angle here.  It is true that VSEPR theory predicts that bonding pairs of electrons (and lone pairs, for that matter) will repel one another as much as possible, thereby increasing bond angles, but this has little to do with bond length.

One way to explain the different bond lengths in $\ce{BF_3}$ and $\ce{BF_4^-}$ is by looking at the molecular orbital diagram of $\ce{BF_3}$:



The linked diagram isn't great since it does not show the filled orbitals: the $1a_2'$ $1e''$ and $4e'$ orbitals are all filled, making the lowest unoccupied orbital (LUMO) the $2a_2''$.  If you haven't wrapped your head around molecular orbital diagrams of this complexity, don't sweat it.  The bottom line is, the $2a_2''$ orbital has antibodning interactions with the $2p_z$ orbitals of boron and the $2p_z$ orbitals of the surrounding fluorines.  Thus, adding electrons to this orbital makes the $\ce{B-F}$ bonds weaker, and therefore longer.