I wanted to know the origin and significance of the prayer called Salāt al-Tasbīh.
Salāt al-Tasbīh, it should be appreciated, is not a practice initiated by the Holy Prophet (sws) in the Ummah as a Sunnah. Consequently, we see that it has only been mentioned in some individual reports. Moreover, most famous and widely acknowledged books of H~adīth like Bukhārī and Muslim and some others do not contain this report.
Thus we cannot be sure of the reliably of these reports beyond any doubt. The importance of the prayer (regarding the stress it received) can obviously be ascertained by the fact that it is only mentioned in some individual reports (these reports mostly rely on two basic chain of reporters) of relatively less reliable works of H~adīth.
According to these narratives whoever offers Salāt al-Tasbīh, his previous sins will be forgiven. The Holy Prophet (sws) is reported to have said to Abbās, his uncle, to offer it very often. He said that one must offer it once in a lifetime at least. However, this exhortation does not raise the importance of the prayer in religious obligations. Had the Holy Prophet (sws) intended to include it in the Sunnah he would not have confined this information to only some of the individuals rather he would have disseminated it among the entire generation.