Picture is sometimes worth thousand words, so let me share one with you. Below you can see illustration that comes from Bradley Efron's (1977) paper [*Stein's paradox in statistics*][1]. As you can see, what Stein's estimator does, is that it moves each of the values closer to the grand average. It makes values greater then grand average smaller and values smaller then grand average, greater. By shrinkage we mean *moving the values toward average*, or *towards zero* in some cases like regularized regression that shrinks the parameters towards zero.

[![Illustration of Stein estimator from Efron (1977)][2]][2]

Of course, it is not only about shrinking itself, but what [Stein (1956)][3] and [James and Stein (1961)][4] have proved, is that Stein's estimator dominates maximum likelihood estimator in terms of total squared error,

$$
E_\mu(\| \boldsymbol{\hat\mu}^{JS} - \boldsymbol{\mu} \|^2) < E_\mu(\| \boldsymbol{\hat\mu}^{MLE} - \boldsymbol{\mu} \|^2)
$$

where $\boldsymbol{\mu} = (\mu_1,\mu_2,\dots,\mu_p)'$, $\hat\mu^{JS}_i$ is the Stein's estimator and $\hat\mu^{MLE}_i = x_i$, where both estimators are estimated on the $x_1,x_2,\dots,x_p$ sample. The proofs are given in the original papers and the appendix of the paper you refer to. In plain English, what they have shown, is that if you make simultaneously $p > 2$ guesses, then in terms of total squared error, you'd do better by shrinking them, as compared to sticking to you initial guesses.

Finally, Stein's estimator is certainly not the only estimator that gives the shrinkage effect. For other examples, you can check [this blog entry][5], or referred *Bayesian data analysis* book by Gelman et al. You can also check the threads about regularized regression, e.g. https://stats.stackexchange.com/questions/20295/what-problem-do-shrinkage-methods-solve, or https://stats.stackexchange.com/questions/4272/when-to-use-regularization-methods-for-regression, for other practical application of this effect.


  [1]: http://statweb.stanford.edu/~ckirby/brad/other/Article1977.pdf
  [2]: https://i.sstatic.net/FXtoT.png
  [3]: https://projecteuclid.org/euclid.bsmsp/1200501656
  [4]: https://projecteuclid.org/euclid.bsmsp/1200512173
  [5]: http://andrewgelman.com/2014/01/21/everything-need-know-bayesian-statistics-learned-eight-schools/