How did Tolkien influence later writers?
A few thoughts...
Tolkien broke several conventions in his books, notably in Lord of the Rings, originally published in three volumes, with a complex back story and rich setting. Tolkien's academic interest in the historical development of languages, his personal exeriences of the First and Second World Wars, his poor health, all contributed to an understanding of culture and heroic struggles that comes over in his writing.
I think modern fantasy writers have learned from Tokien's example and adopted some of it:
Middle Earth or something like it - its creatures, races, languages and technological level - can be adopted as a ready made backdrop for a fantasy story.
Having a rich back story or collection of stories gives an impression of depth and significance.
Characters can be complex, with mixed motives - just like you and me. Consider Boromir for example.
Multi-volume stories can work (and create a saleable brand).