Seeing as multiplayer is integral to the Arms gameplay experience -- with Nintendo flaunting it as an eSports contender during E3 -- the introduction of a LAN connection should help with competitive gaming down the line, particularly during major tournaments where stability and reaction times are critical.
The other update is slightly less exciting. "Arena Mode" basically lets up to four players join a lobby, allowing the players that aren't trying to beat down one another to watch as spectators. You'll be able to pick from four viewpoints whilst inactive, including a roving camera that follows the multiplayer battles. Nintendo's take is like a very small-scale version of Twitch with an audience of just... two.
Nintendo has a lot riding on the success of Arms. Aside from the competitive gaming aspirations, it also has to sustain interest in the Switch, which launched with just a handful of titles, including the much celebrated Legend of Zelda: Breath of the Wild. Thus far, it's met with success in its native Japan, where Arms is estimated to have debuted atop the sales charts, but has failed to match that feat in the UK, instead reportedly settling for second place behind Horizon: Zero Dawn upon its release.