The second stanza is full of clues that point towards the death of the athlete, even if we did not have the title to point it out to us. Note what the speaker says to the athlete who has died in the second stanza:
Today, the road all runners come,
Shoulder-high we bring you home,
And set you at your threshold down,
Townsman of a stiller town.
Note that this road is one that "all runners come," pointing towards the eventual death of everyone, no matter how great and famous. Likewise, being set down at your threshold points towards being placed in your grave, while the town we are told is still as it unites in mourning the loss of one who received such adulation and praise.
Note the way in which both first and second stanzas repeat that the athlete is carried "shoulder high" by the crowd, which serves to reinforce the ironic parallel between the two processions as the first remembers his moment of glory, and the second recalls his moment of death.