Could someone teach me why the imperfect subjunctive tense is being used in the sentences like these?
Can I just use the present tense?

"Me gustaría que ganáramos la competición atlética. "
"Solo faltaría que las dos cooperaran mutuamente, y entonces ya no habría nada qué decir."

See a translation

Yes you can.

Me gustaría que ganemos la competición atlética.

Solo faltaría que las dos cooperen, y entonces ya no habría nada que decir.

Functionally, the tense doesn't really make that much of a difference in this context. If I had to force one I would say the ones present could sound slightly more like a thinly veiled request in the proper context (like "hey, let's win this" or "hey, cooperate"), but again, that's a bit of a stretch and the ones in imperfect could also be interpreted in that way. I do however feel that the imperfect tense kind of softens things up a bit, if that makes sense.