You need to do this because you were using a string in onSubmit event.
<form className='commentForm' onSubmit={this.handleSubmit}>
You had this in your sample code:
<form className='commentForm' onSubmit='handleSubmit'>
Your code caused a Uncaught TypeError: string is not a function
error. Because of that error it was not hitting the handleSubmit
function and also caused the browser to reload.