1To be able to speak or talk (a language); to have the ability to converse or to be allowed/able to speak.
私は日本語が少し話せるようになった。
I've become able to speak Japanese a little.
彼女は英語でなら話せるが、専門用語は苦手だ。
She can speak in English, but she's not good with technical terms.
この件については上司となら話せる。
I can talk about this matter with my boss.
2To be understanding or reasonable; someone you can talk to—able to listen and respond sensibly.
あの先生は話せる人だから、困ったことがあれば相談してみて。
That teacher is someone you can talk to, so consult them if you have a problem.
彼は冷静で話せる性格だから、感情的にならずに話し合える。
He's calm and reasonable, so you can discuss things with him without emotions flaring.