Offense is the core appeal to many of these games to newcomers. Who doesn't want to wail on their opponent with magic fireballs and wicked punches? As the neutral game develops at the start of each round, it's always a race to who gets the first hit. Landing that hit means it's go time on offense.
In more technical terms, offense refers to the player state where landing a hit or assuming positioning puts the opponent into a defensive stance lest they absolutely explode from a starter hit.
Some character toolkits have strong offense from their ridiculous mobility options, or maybe their ability to mix up the correct defensive option for that situation, or even just existing on screen from how hard they hit.