There is a phenomenon known as p(pi)-p(pi) back bonding taking place in BF3. It may sound a bit complicated, but i can explain in simple terms.
now, when Boron forms bonds with fluorine, boron has 6 electrons and fluorine has 8 electrons. Now fluorine being small in Size and having High Electronegativity (tendency to pull negative charge) has a high amount of charge density (charge per volume) on it, which it prefers to reduce. So, it loses an electron and Boron accepts that electron. Then Fluorine makes a bond with the electron it just lost, forming a double bond. This is known as p(pi)-p(pi) back bonding. As a result of this, both boron and fluorine have eight electrons. All three fluorine atoms are capable of this type of bonding, making boron a highly stable molecule.