The typical olefin polymerization reaction starts with a metal dimethyl species, #L_2M(CH_3)_2#, where #(i)# #M# is a metal such as titanium or zirconium, and #(ii)# #L# is some ligand that stabilizes and solubilizes the metal centre.
Addition of a Lewis acid abstracts one of the methyl groups to yield a species conceived to be #L_2M^(+)(CH_3)#:
#L_2M(CH_3)_2 + "aluminum reagent"rarrL_2M^(+)(CH_3) + "methylated aluminium reagent"#
Ethylene can bind to the cationic metal centre:
#L_2M^(+)(CH_3)+H_2"C=C"H_2rarrL_2M^(+)(CH_3)(eta^2-H_2"C=C"H_2)#
The methyl group bound to the metal centre can migrate to the #eta^2-"bound"# olefin to give a propyl group. And such insertion generates an open coordination site on the metal centre, to which another ethylene can bind, and this reinserts into the #M-"alkyl"# residue. Long chain alkyl residues, polyethylenes, can be built up this way.
This will be much better explained in your given text.