The clue here is in the verb. When the subject has ने, the verb doesn't agree with it - it usually agrees with the direct object. Since खेल isn't showing number on its own, the verb form is the only thing showing the number.
हमने खेल खेला -- The verb khelaa is singular, so the object must be interpreted as singular, so "We played a/the game"
हमने खेल खेले -- The verb khele is plural, so the object must be interpreted as plural, so "We played (the) games"
Compare this with something where you can see the number on the object:
हमने कुत्ता देखा -- 'kutta' is singular, so the verb is agreeing as singular (and masculine) 'dekhaa'
हमने कुत्ते देखे -- 'kutte' is plural, so the verb is agreeing as the masculine plural now - 'dekhe'
So comparing it with this sentence, हमने खेल खेले, even though there's no overt plural marker on the object, the verb has to keep track of whether the object is plural or not, and that dictates the ending on the verb. Hope that's helpful!
Where do you place words like कल or आज in a sentence? Also, where should I place question words like क्यों, क्या or कब in a sentence. e.g how would I translate the sentence, 'When do you want to play games?' or 'Why did you eat my roti?'
for the words "कल" or "आज" you could usually put them either right at the beginning of the sentence or just after the subject (in this case the subject is we - "हमने") so the sentence could also be "हमने कल खेल खेले।"
for the question words, they are typically placed right before the verb. so if you wanted to say "Why did you eat my roti?" it would be "आपने मेरी रोटी क्यों खाई?" in this case, "खाई" is the verb (ate) and the question word "why" is right before it. for "When do you want to play games? it would be "आप खेल कब खेलना चाहिए?", in this case, "खेलना" is the verb (to play) and the question word "when" is right before it.
Hope this helps! :)