The quantum phase transitions induced by a magnetic field are theoretically studied in a frustrated two-leg spin ladder. Using the density-matrix renormalization-group method, we find some magnetic phase transitions and plateaus in two different cases of strong and weak rung couplings. With the strong rung coupling, the three magnetization plateaus are found at 1/3, 1/2, and 2/3 due to the frustration. Those can be understood in terms of a quasispinon reconstructed from the singlet and the triplets of spins on a rung. The plateau at 1/2 corresponds to the valence bond solid of the quasispinons, while the plateaus at 1/3 and 2/3 can be associated with the array of quasispinons such as the soliton lattice. This is different from the usual BoseEinstein-condensation picture of triplons. Our results will be useful in understanding magnetization curves in BiCu 2 PO 6 .