Special teams is the collection of plays built around kicks: punts, field goals, extra points, kickoffs and returns. The exact NFL kickoff format has changed over time, but the basic strategic purpose remains simple: points and field position.
Punts
A punt normally happens on fourth down when the offence decides that trying to convert is too risky. The punter gives up possession in exchange for pushing the opponent farther away from the end zone.
Field goals
A field goal is worth three points. The farther the attempt, the harder the kick and the more field position a miss may surrender.
Returns
Return teams are trying to create hidden yardage before the offence even takes the field. Coverage teams are trying to prevent that and force a long field.
Why special teams changes offensive strategy
A great punt may pin an opponent near its own goal line. That changes the play-calling menu because the offence has less margin for error. A strong return can do the opposite and shorten the field.
Simple way to think about it: offence tries to gain yards, defence tries to stop yards, special teams often decides where those yards have to start.