It's the polite form of "kikaeru" (聞かせる) and is used when speaking to someone of higher status or someone you want to show respect to.
Here's how the meaning can vary depending on context:
* "To let someone hear": This meaning is used when you want someone to listen to something, whether it's music, a story, or just a sound.
* Example: "Ano uta o kikasemasu." (あの歌を聞かせます。) - "I'll let you hear that song."
* "To play (music)": This meaning is used when you are playing music for someone to listen to.
* Example: "Kore o kikasemasu." (これ を 聞かせます。) - "I'll play this for you."
* "To tell (a story)": This meaning is used when you are telling someone a story.
* Example: "Omoshiroi hanashi o kikasemasu." (おもしろい話を聞かせます。) - "I'll tell you an interesting story."
Remember that the nuance of "kikasemasu" can depend on the situation and the context in which it's used.