Why is BIC considered consistent (though AIC is mostly used) for large number of observation? AIC, BIC are the famous criteria for model selection. But many times they show different results.
I read in several places that BIC is consistent while AIC is not. And AIC can achieve minimax rate but BIC can not.
However I do not actually get the concept behind these. An intuitive and lucid explanation is appreciated.

Perhaps Chapter 4 of Claeskens & Hjort "Model Selection and Model Averaging" (2008) or Claeskens "Statistical Model Choice" (2016) may help.
The former reference is more detailed and perhaps clearer. I am not going to reproduce it here and would recommend reading the original. The latter is briefer and contains some relevant quotes:
Consistency:

Usually one investigates consistency or efficiency of model selection criteria. Sin & White (1996) distinguish between weak and strong consistency and between situations where there is clearly one model with a best value of the information criterion and other situations where the criterion values are in some sense close and one hence prefers to select the more parsimonious model, the one corresponding to the smaller penalty. Under some other conditions on the models, if the penalty grows with the sample size in such a way that the penalty divided by the sample size still converges to zero, the criterion is consistent. The BIC is such a consistent criterion. When the sample size tends towards inﬁnity a consistent criterion is able to select the more parsimonious model when two models have the same best value of the criterion. When the true model is part of the model collection, a consistent criterion is able to asymptotically select this true model.

Efficiency:

A criterion such as AIC with a penalty that does not grow with the sample size, rather overﬁts, it does not necessarily pick the simplest model. For prediction purposes, this might be advantageous while the opposite behavior of underﬁtting might cause more harm to predictions. This is an intuitive reasoning behind efficiency. A model selected with an efficient criterion behaves as well as the best model in the model collection in terms of mean squared prediction error when the sample size tends to inﬁnity. The criterion AIC is efficient (Shibata, 1980), BIC is strongly consistent.

Consistency vs. efficiency:

Yang (2005) has shown that efficiency and consistency cannot take place at the same time. Hence, adding the penalties of AIC and BIC, for instance, does not make a better criterion.

(Emphasis in italics is the author's. Emphasis in bold is mine.)
