Think of the one who takes care of whatever is left. Sweep is the action and -er is the one who does it, and because -er covers both people and machines, a sweeper can be a worker with a broom or a truck with brushes underneath. Hold on to the idea of what is left over and the football use follows: the player behind everyone else, dealing with whatever got past the rest.
From the word’s history
Wiktionary records sweeper from Middle English swepere, equivalent to sweep + -er, with a Middle Low German cognate used for a broom. The -er that covers both a person and a tool is there from the start.
字典記載 sweeper 來自中古英文 swepere，也就是 sweep ＋ -er，並提到中古低地德文的同源字指的是掃除用的掃帚。「人和工具共用同一個 -er」這件事從一開始就在，所以今天它既能指人，也能指機器。
Wiktionary entry for sweeper, packet etymology_text: "From Middle English swepere, swepare, equivalent to sweep + -er."
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