In The Intellectuals and the Masses, John Carey describes Eliot as a prime example of the modernist elite who wrote self-consciously difficult work which excludes the common reader. Allusions are a central part of this process, since the educated audience to which Eliot is addressing himself will recognize them, whereas the ordinary people will be lost.
This is an uncharitable explanation of Eliot's use of allusion in The Waste Land, though probably one that has occurred to most readers in their attempts to make sense of the poem. However, there is a related but less pejorative explanation which seems to align with Eliot's concerns in the poem. The allusions continually take the reader away from the squalor of contemporary London to the splendor of ancient civilizations, often as described by the greatest of poets. This provides both variety and contrast, allowing some relief from the gloom without lessening Eliot's condemnation of the modern city, which appears in an unflattering context.
To add another level of complexity, Eliot often adulterates the allusions with modern elements, as though nothing touched by modernity can remain pure. This technique is perhaps most in evidence at the beginning of the second section, "A Game of Chess," when Shakespeare's famous description of Cleopatra degenerates into a description of something like the jewelry and perfume departments of a large, opulent store.