How do American football officials measure first down yardage with chains, especially when they go on the field to confirm first downs?

The first down marker is traditionally two orange poles with a 10 yard chain connecting them (10 yards from the inside edge of one pole to the inside edge of the other pole). Once a ball is spotted for play on 1st down, the Linesman spots his foot on the line of scrimmage. The chain operator places one pole at the location. Another chain operator places a marker on the multiple of five yard line nearest the end of the chain. If the play results in a spot close to the line to gain (10 yards from teh 1st down line of scrimmage), then the marker is placed on the multiple of five yard line and the chain extanded to the spot of the ball. If the nose of the ball touches or goes past the iside edge of the pole, it is a first down.

https://en.wikipedia.org/wiki/Down_(American_football)