A proposed laser-plasma-based relativistic electron source [E. Esarey et al., Phys. Rev. Lett. 79, 2682 (1997)] using laser-triggered injection of electrons is investigated. The source generates ultrashort electron bunches by dephasing and trapping background plasma electrons undergoing fluid oscillations in an excited plasma wake. The plasma electrons are dephased by colliding two counterpropagating laser pulses which generate a slow phase velocity beat wave. Laser pulse intensity thresholds for trapping and the optimal wake phase for injection are calculated. Numerical simulations of test particles, with prescribed plasma and laser fields, are used to verify analytic predictions and to study the longitudinal and transverse dynamics of the trapped plasma electrons. Simulations indicate that the colliding laser pulse injection scheme has the capability to produce relativistic femtosecond electron bunches with fractional energy spread of order a few percent and normalized transverse emittance less than 1 mm mrad using 1 TW injection laser pulses.