Boron atom has an incomplete octet. It can accept a lone pair from X given in the question and can also donate electron to it, thus it can form a π-bond also known as back bonding.
The extent of back bonding depends upon the electronegativity of X and on the size of the valence shell. Lesser is the electronegativity of X and similar is the size of valence shell of X and B, more will be the extend of back bonding. Cl and F are more electronegative than OMe and NMe group, so there extent of back bonding will be least. Among BCl3 and BF3,Cl will have least tendency to form π - bond with boron, because of large size of its 3p-orbital.
Among B(OMe)3 and B(NMe2)3,(NMe) will have the maximum tendency to form π-bond because N in NMe is less electronegative than O in OMe group.
Thus, the tendency to form π -bond with boron follows the order.
BCl3<BF3<B(OMe)3<B(NMe2)3.