We present the analysis of long-slit optical spectra taken of five high-redshift (z almost-equal-to 2-3) radio-loud quasars (QSRs), selected from the imaging survey of Heckman et al. We confirm that the QSRs have Ly-alpha emission that is spatially extended by at least several tens of kiloparsecs, with a luminosity of order 10(44) ergs s-1. The gas is very active kinematically. After a subtraction of the seeing-scattered light from the broad-line region (BLR) of the central QSR, we find that line widths in the nebulae are 1000-1500 km s-1 (FWHM). The gas does not show any globally organized velocity gradients (i.e., velocities do not change by more than about 500 km s-1 over scales of tens of kiloparsecs). The line widths are close to the maximum possible for gas that is freely falling into the potential well of a very massive galaxy. Alternatively, we may be witnessing the explosive ejection of matter on a galactic scale by the QSR. The kinematic properties of the QSR nebulae are broadly similar to those of the nebulae associated with high-redshift radio galaxies. We confirm the suggestion of Foltz et al. that the nuclear He II lambda-1640 emission line in QSRs is systematically narrower than other nuclear emission lines. We find that this narrow-lined, nuclear He II emission has a strength that correlates with the strength of the spatially extended Ly-alpha emission, and suggest that the former probably arises in the inner (arcsec-scale) part of the Ly-alpha nebula. This is consistent with the fact that the velocity of the spatially extended Ly-alpha emission agrees well with the velocity of the narrow core of the nuclear Ly-alpha profile. Selecting by the presence of strong narrow He II lambda-1640 lines should be a good way to find high-z quasars with prominent Ly-alpha nebulae. We discuss the implications of these results for the nature of the z(abs) almost-equal-to z(em) systems in high-z QSRs. In three QSRs, the spatially extended Ly-alpha is redshifted by of order 10(3) km s-1 with respect to the C IV lambda-1549 BLR profile (but exhibits no significant shift in the other two QSRs). We believe that this reflects a blueshift of the C IV lines with respect to QSR systemic velocity. This has implications for the interpretation of the BLR dynamics. We have likely (4-5 sigma) detections of spatially extended He II lambda-1640 in one QSR nebulae and C IV lambda-1549 in another (each at 7%-10% of Ly-alpha). These possible detections and the upper limits on other lines are consistent with the emission-line spectra of high-z radio galaxies. Our data are not good enough either to yield strong constraints on the metallicities in the nebulae or to discriminate decisively between alternative ionization sources (photoionization by a standard active galactic nucleus continuum, photoionization by hot stars, shock heating, etc). The probable He II detector is inconsistent with predictions of photoionization models for normal high-mass stars. The probable C IV detection means that at least some galaxy-scale sites in the early universe have nonprimordial chemical abundances. The probable detections of He II and C IV make it unlikely that the bulk of the spatially extended Ly-alpha emission arises via resonant scattering of nuclear Ly-alpha photons by neutral hydrogen surrounding the QSR.