Scientometrics is a science to the extent that it applies the scientific method to a field of inquiry. Researchers in this area formulate questions and conceptualize existing problems (e.g., “in these times of scarcity, the public wants research to be efficient: how can we measure this?”), they make hypotheses, make predictions, gather data to test them, analyze the data and prove or disprove their hypotheses.
Regarding “what do we scientifically learn from scientometrics?”, well, there are plenty of established results (you can find some on wikipedia), but I'll use one recent paper from Scientometrics which highlights what this field can bring us:
Negative results are disappearing from most disciplines and countries
D. Fanelli, Scientometrics 2012, 90, 891–904
This one was really an eye-opener for me, on a phenomenon which I always supposed existed, but it was nice to see it backed by hard data. Other examples include:
Physical and economic bias in climate change research: a scientometric study of IPCC Third Assessment Report
A. Bjurström and M. Polk, Climatic Change 2011, 108, 1–22
Language biases in the coverage of the Science Citation Index and its consequences for international comparisons of national research performance
T. N. van Leeuwen, H. F. Moed, R. J. W. Tijssen, M. S. Visser, A. F. J. van Raan, Scientometrics 2001, 51, 335–346
I'll finish with a personal opinion: while I am annoyed, as most people, with the emphasis currently given on bibliometrics in the evaluation of research and research projects, I think scientometrics in general has an important role to play, just like related fields such as studies on ethics of research: better understanding the positive and negative implications of the way we currently do science is healthy (the meta-level of research).