Here are three examples from combinaorics:
1) The Frankl Wilson' theorem (The paper can be found here). This theorem in extremal combinatorics has a large number of amazing applications: Explicit Ramsey constructions, applications in combinatorial geometry; applications regarding Shannon capacity of union of graphs and many more.
2) Trotter-Szemeredi The result by Trotter and Szemeredi regarding the maximum number of incidences between points and lines in the plane had remarkable applications including one discovered by Elekes' to the product-sum theorem.
3) The mod p product sum theorem by Bourgain-Katz-Tao had many surprising applications in many directions. (One reason for the wide applicability is that when you multiply matrices sums and products mix.)