解(jiě)释(shì) can be translated into ‘explain’ in English, which means account for the information that the listener has already received(By seeing, hearing or other vias, but probably wrong) more clear. For example,
(1)你(nǐ)误(wù)会(huì)我(wǒ)了(le)，请(qǐng)允(yǔn)许(xǔ)我(wǒ)解(jiě)释(shì)。
You misunderstood me, please allow me to explain.
说(shuō)明(míng) can be translated into ‘introduce’ in English, which means provide the new information that the listener are familiar with. For example,
(2)这(zhè)件(jiàn)事(shì)大(dà)家(jiā)不(bù)知(zhī)道(dào)，我(wǒ)来(lái)说(shuō)明(míng)一(yí)下(xià)情(qíng)况(kuàng)。
You don’t know this matter, let me introduce the situation.
