Translation:I think we will play soccer today.
Jack's point is that almost exclusively in the US (and, thanks to the US, in Japan) do people say "soccer". To the rest of the world that same sport is "football" (since the US kind is "American football", even in Japan!). Making "soccer" the only acceptable answer may not be technically wrong, but it's pretty inconsiderate.
To accept "I'm thinking" would require changing the question to the -て form, as in "思っています" instead of what they used, "思います."
So with how they worded the question, it is just Present Tense, "I think" rather than the Present Continuous Tense of "I am thinking."
The question would have to be "今日はサッカーをやると思っています"
You know, you should read a question carefully before answering anything. ;) This person asked about the use of と in this sentence, and you answer with a completely different one...
So, here と is just marking the quote "I think ~" => "~と思います". It is used with verb of declaration like 言う. Hope it will help you :)
I don't understand why "I think we play soccer today" is not accepted? I mean, present tense or future one is the same in Japanese, and today is the main subject of the sentence so it could something like : "What do we do in sports class today? - I think today we play soccer"
I may be wrong since I'm not a native so feel free to correct me! Thank you.
The are not opposites. They are two different verbs meaning different things. やる (usually written in kana) means 'do', 'play', 'perform' and やめる (止める) means 'stop', 'resign', 'give up' depending on the context of the sentence (if the meaning is 'resign' or 'quit', a different kanji would be used: 辞める). I hope this helps.