What measurements and observations of the Hulse-Taylor Pulsar agreed precisely with the predictions of general relativity, and how did these measurements support the theory?
One of the predictions of General Relativity is that certain objects can give off energy in the form of gravitational radiation. This means that over time, the orbit of the two neutron stars should "decay", and they should come closer to each other. Using General Relativity, it is possible to predict the changes in the orbits and the energy of the emitted gravitational waves; analysis of the system showed that General Relativity's predictions were correct. More information (certainly more in-depth) can be found here and here.
N.B. Studying gravitational waves requires linear approximations of the theory of Relativity, involving mathematics that I can't quite explain properly but, fortunately, aren't too important to this topic.
I haven't been able to find out if other theories (such as Brans-Dicke theory) make similar predictions that agree with observations.
Referring to the past edit: See this paper for details on the differences in predictions. Long story short, Brans-Dicke theory and general relativity may make identical predictions of gravitational waves, but there is no definite answer.