If most of jet streams are westerly, that is to say, they blow from west to east, why does the tropical/equatorial jet stream blow from east to west?
Looking at the annual climatology of the 200 hPa winds as shown below from NCAR NCEP reanalysis it is noted that the mid latitudes have by and large westerly winds (for a planet that rotates west to east). On the other hand averaged over a latitude circle(zonal average) the low latitude tropics and equator have by and large easterly winds. The reference Annual cycle of equatorial east-west circulation over the Indian and Pacific Oceans argues that tropical upper level divergence over the Central Pacific is the reason for the upper level equatorial easterlies present over the Indian Ocean and West Pacific Ocean. On the other hand annual mean upper level equatorial winds over the East Pacific and east Atlantic are by and large westerlies(if one were to perform zonal regional averaging). The reason is beyond the scope of this answer and is due to mid latitude origins.
The upper level equatorial easterlies never quite approach the definition of a "jet" on annual scales except during the boreal summer period. So one needs to look at seasonal scales to explain the formation of the so called Tropical Easterly Jet Stream.
From these two pioneering papers Koteswaram TEJ and Raghavan K TEJ 1973 (available behind a paywall, reproduced here - Origin of Tropical Easterly Jet), there exists at the upper levels an anticyclone over the Indian subcontinent, also known as the Tibetan anticyclone. The Tropical Easterly Jet(TEJ) is believed to be connected with the Tibetan anticyclone. The Tibetan anticyclone is present in the upper troposphere (100 to 150 hPa).
In the above image derived from NCEP NCAR reanalysis data the June through September climatology of the 100 hPa geopotential height is presented and one can clearly see the Tibetan anticyclone as part of the broader monsoon anticyclone.
So the location of the Tibetan anticyclone is in accordance with seasonal shifts. In winter the Tibetan anticyclone is further south (around 10 N) and during summer it moves to warmer latitudes.
Lower levels of the anticyclone is tilted equatorward during the boreal summer and poleward during the winter. However the upper levels of the anticyclone are tilted poleward during all seasons.
Koteswaran and Raghavan have calculated the zonal component of the equatorward outflow from the anticyclone and there appears to be no comparison with the upper level easterlies associated with the Indian Summer Monsoon(ISM). But there appears to be strong comparison with the winds derived from thermal gradient over the Tibetan Plateau. So it appears that the winds are originating due to thermal reasons alone. The TEJ is to a large extent influenced by the temperature distribution in the upper levels as a result of vertical motion associated with ISM. The Tibetan plateau comes under the influence of both sensible and latent heat and leading to the formation of a surface heat low
Shown here is the sea level pressure climatology of the June-September period and one can clearly see the heat low over the elevated Tibetan plateau.
Given the climatology of the observed surface pressure(this surface pressure has NOT been extrapolated to sea level pressure) during the June through September period this dataindicates that the mean value is around 500 hPa it might be helpful to look at the temperature of the 500 hPa climatology.
Why should one want to look at the 500 hPa temperature surface ? it is because 500 hPa is the average surface pressure over the Tibetan Plateau(which is an elevated plateau). Normally the 500 hPa pressure surface over sea levels is found at a height of 5 kms. Once again one can clearly see from the 500 hPa temperature surface that the Tibetan plateau provides for an elevated heat source.
Krishnamurti in the book Monsoon Models offers a detailed explanation
the southerly outflow of the subtropical jet provides a mechanism where eddy kinetic energy is converted into kinetic energy of the mean flow and all this happens above 250 hPa. Stratospheric easterly winds penetrate downwards into the northern branch of the TEJ. Above all the conversion of available potential energy into kinetic energy above 250 hPa in the region of increasing pressure gradient caused by the proximity of the Tibetan anticyclone as well as the Coriolis force convert the diabatic heat into kinetic energy of the TEJ.
The above two images provide a visual representation of the explanation provided in the quoted text i.e. southerly outflow of subtropical jet + Coriolis force.
In recent years due to the decrease in the temperature contrast between the land and sea over the Indian subcontinent the TEJ has shown a decreasing trend and there are several studies that illustrate this trend as well.