Wittgenstein and his philosophy 

Wittgenstein was considered by some to be the greatest philosopher of the twentieth century. His main concern was the the philosophy of language. To understand Wittgenstein, we need to know how we learn language.

The first method, the hypothetical one, is to learn by checking dictionary. Each word is defined by other words, which in turn are defined by other words. After several rounds, you will find some of the defining words were being defined earlier. This is a circular definition. Nothing is truly defined.

Fortunately, this is not how we learn our mother tongue. We learn the first words from our interactions with our family and friends. It is only later we occasionally pick up a dictionary.

But early researchers on the philosophy of mathematics, languages, and mind attempted to establish the foundation on logic, that is, the dictionary style of learning. The most prominent researchers among them were Bertrand Russell and Ludwig Wittgenstein. (The following passages are mainly copied from wikipedia.)

Wittgenstein’s philosophy is often divided into an early period, exemplified by the Tractatus, and a later period, articulated primarily in the Philosophical Investigations. The "early Wittgenstein" was concerned with the logical relationship between propositions and the world, and he believed that by providing an account of the logic underlying this relationship, he had solved all philosophical problems. The "later Wittgenstein", however, rejected many of the assumptions of the Tractatus, arguing that the meaning of words is best understood as their use within given contexts.

According to Wittgenstein, philosophical problems arise when language is forced from its proper home into a metaphysical environment, where all the familiar and necessary landmarks and contextual clues are removed. He describes this metaphysical environment as like being on frictionless ice: where the conditions are apparently perfect for a philosophically and logically perfect language, all philosophical problems can be solved without the muddying effects of everyday contexts; but where, precisely because of the lack of friction, language can in fact do no work at all. Wittgenstein argues that philosophers must leave the frictionless ice and return to the "rough ground" of ordinary language in use. Much of the Investigations consists of examples of how the first false steps can be avoided, so that philosophical problems are dissolved, rather than solved: "The clarity we are aiming at is indeed complete clarity. But this simply means that the philosophical problems should completely disappear."

The "later Wittgenstein" effectively rejected the significance of the work of “early Wittgenstein”. 

How about the significance of the work by “later Wittgenstein”? Bertrand Russell made the following assessment.

I have not found in Wittgenstein's Philosophical Investigations anything that seemed to me interesting and I do not understand why a whole school finds important wisdom in its pages. Psychologically this is surprising. The earlier Wittgenstein, whom I knew intimately, was a man addicted to passionately intense thinking, profoundly aware of difficult problems of which I, like him, felt the importance, and possessed (or at least so I thought) of true philosophical genius. The later Wittgenstein, on the contrary, seems to have grown tired of serious thinking and to have invented a doctrine which would make such an activity unnecessary. I do not for one moment believe that the doctrine which has these lazy consequences is true. I realize, however, that I have an overpoweringly strong bias against it, for, if it is true, philosophy is, at best, a slight help to lexicographers, and at worst, an idle tea-table amusement.

Russell’s reaction was natural. By rejecting his own early work, Wittgenstein also rejected Russell’s efforts. Russell’s worry was also justified. Modern philosophy is, at best, a slight help to lexicographers, and at worst, an idle tea-table amusement.

If so, why is Wittgenstein so popular?

Argentine-Canadian philosopher Mario Bunge considers that "Wittgenstein is popular because he is trivial."In Bunge's opinion, Wittgenstein's philosophy is trivial because it deals with unimportant problems and ignores science. According to Bunge, Wittgenstein's philosophy of language is shallow because it ignores scientific linguistics.  Bunge also considers Wittgenstein's philosophy of mind to be speculative because it is not informed by the scientific research performed in psychology.