The sentence actually is presented in two different exercises.
In the exercise you see here in the discussion it says "Warum möchten sie mitmachen?"; in this case "sie" is, as hohenems pointed out, is the plural form and the translation is "why do THEY want to join".
In the exercise I just did, "Listen and write in German" I wrote "Warum möchten Sie mitmachen?" Then. of course, you have the option to use the formal "you" and your answer can be "Why do YOU want to join"
Look at these verb endings:
sie (they) gets -en sie (she) gets -e
OK, so if the sentence can be translated into the present "Why do they want..." then what is the translation of "Warum moegen sie......" and lets suppose you really wanted to emphasize the WOULD as in "Why would they want to join such a terrible organization?" how would you translate that to get the full meaning.
Those are not versions, but different words:
- wollen - to want
- wünschen - to wish
- brauchen - to need
- mögen - to like (subjunctive form "möchten" is translated as "would like")
There is also another word that is used to express the same idea - gern. For example "he would like to read it" can be translated as "er würde es gern lesen".