The English sport comes from the French for the same… desporte — notice it is the same as the Spanish, except with an extra “s” (that’s a pattern that we’ll explain in the French version of this page one day!).
You can see the connection to the English clearly if we remember the “s” and we remember the de- prefix was lost over time. Thus, the s-p-r-t maps to the Spanish (d)-(s)-p-r-t.
The French desporte (and thus the English sport) and its Spanish equivalent deporte both come from the same Latin root: des- meaning “away” and portare, meaning, “to carry”.
Thus deporte, and sport, is also related to puerto (“port”) and portero (“super”, in the sense of, “superintendent”) in Spanish and port in English.