You set up a team and choose other players' teams to "fight" against. Each card's point value will be matched against their opponent's, and the higher one wins. The cards' types work like rock paper scissors - pink gets a triple bonus against green, green gets a triple against purple, and purple gets triple against pink. The first time a card on either side gets defeated, they will be replaced by a helper card that is the middle card on the team of someone on your friend list, and the helper will battle the card that defeated the first. If someone doesn't have a team (usually because they're inactive) then it simply selects the card they have on their main page. Helpers are randomized each time you select an opponent, so you can keep canceling out of a match to roll for a strong helper.
Players with certain point totals are worth more tickets, which you need to spin the box gacha. From the battle page, you can select what card type you want your opponents to focus on, and what rank they are. (Rank 1 is worth 5 tickets, and vice versa) At high rank, getting strong helpers is crucial to victory - if your team focuses on a single element, never accept a helper of the same element as your current team, because chances are good the card that defeated one of your cards was one that was strong against your element. Also, any time someone challenges your team and loses, you get 3 tickets.
If you can defeat 50 opponents of any rank without losing, you'll get one of the event's SRs. The other two SRs are from the gacha box, which contains 400 items ranging from booby prizes to consumables to cards. Once you obtain the SR from the box, you can choose to refill the box to 500 items, but this time with another, different SR in it. There is also an R card earned by winning 200 times in total.