Example sentences
kě可yǐ以zǒu走le了。
You can go.
zhè这yàng样bù不kě可yǐ以。
This isn't allowed.
About this word
可 means may or to approve, and 以 was the classical preposition by means of, so the pair began as a phrase closer to "can use it to do something". That is why it landed on having a way to do a thing. Modern 可以 mostly grants permission (你可以走了, you may go) or points to an option (我们可以坐地铁). This separates it from 能, which is about ability or circumstance, and 会, which is about a skill you have learned. Asking 我可以...? is the standard polite way to request permission.
The story behind the character
可
可 is 口, a mouth, with 丂 at the side giving the sound. Xu Shen wrote 可, 肯也: to be willing, to say yes. A mouth making the sound 丂, and the meaning landed on approval. Can, may and okay all grew out of a yes. 可以 puts 可 with 以, to use, so the phrase reads as yes, let us use this.
以
以 in the oracle bone is a person carrying something, or a tool in use. Xu Shen wrote 以, 用也, to use, and explained the shape as a reversed 巳. That one meaning explains a lot of grammar: 可以, can use it, 所以, that by which. It is one of the busiest characters in the language, and underneath it is still a hand carrying something.