The English plural -s is the only survivor of a much more complicated Old English nominal declension system. As you can see from the tables on the linked Wikipedia article, the plural ending for the Nominative and Accusative of "strong masculine nouns" was -as, and as the Old English nominal system broke down, this ending was generalized to all nouns in all cases. By Middle English we only have the ending -es for all nouns, and in Modern English the -e- has disappeared (except in spelling in some cases), giving us the plural -s.
The verbal ending -s for the third-person singular in the present tense comes from someplace completely different. As late as Early Modern English (the King James Bible, Shakespeare, etc.) the 3sg ending was -eth, which in turn goes back to an Old English ending -eþ (though as above, the full Old English system was far more complex than anything in Middle English or Modern English). A sound changed turned the ending -eth into -es, and the vowel was lost by the same process that eliminated the vowel from -es in the plural.
Thus it's mostly a perverse coincidence that the ending for plural nouns is also used with singular verbs. The two endings have completely different origins, and now look alike because sound changes turned both of them into the same thing.