Short rate models were first used in the 1970s and 1980s to try to fit and explain the term structure of interest rates - they went beyond simple parametric shapes (polynomials and exponential forms). They were not used for pricing as the fact that these short-rate models (Vasicek, CIR and Ho-Lee) had only two or three free parameters meant that they could not exactly fit the term structure of interest rates.
This lack of fit was not seen as a problem by their early users because what a short-rate model gave was a way to capture the relationship between the shape of the term structure of interest rates and the term structure of interest rate volatility. This was important if you wanted to understand the value of bond convexity.
A multifactor version of such a model might also be argued to have some "economic" properties that might make one think that it captures the relationship between various points on the curve and deviation from this curve may be considered to be a "mispricing". In the early 1990s, hedge funds such as LTCM used multifactor extensions of these models in this way and used them to put on massive "convergence trades". Some people still use them like this.
Concerning their use in derivative pricing, in the early 1990s, caps and floors and European style swaptions could all be priced using Black's model. However, for more exotic products, and also for these products, a more complete model was required. However these short-rate models did not refit the initial term structure of interest rates and so could not be used.
This problem was solved by Heath, Jarrow and Morton in the late eighties (published 1990) who showed how to construct a drift that would ensure a fit to the initial term structure of interest rates making it arbitrage-free. Although HJM is based on a forward curve, it is also possible to apply this to a short-rate model. So Hull and White showed how to do this to Vasicek's short-rate model, and others showed how to do this to other rate processes such as Black-Derman-Toy and Black Karasinski.
Although forward rate models such as BGM are now quite dominant, arbitrage-free short rate models still play a large role in derivative pricing. HW is still popular due to it having a fast analytical solution to the bond price. HW and others are also used for multi-callable products as they can be more easily implemented on binomial and trinomial trees than forward rate models which rely more on Monte Carlo techniques.