A "chance word" is something heard by luck or happenstance
Let's look at two definitions from Merriam-Webster.
something that happens unpredictably without discernible human intention or observable cause
And then a definition of "by chance":
in the haphazard course of events
Here's a third defintion, this time the 2a one for "word"
a brief remark or conversation
Putting these together, a "chance word" is a "brief remark" heard by chance, or "unpredicatably", without "intention or observable cause".
Imagine just walking around randomly, and you hear someone saying "I love puppies". There was no intention for you to hear this; it was heard unpredicatably. It was a "brief remark". Therefore this was a "chance word".
The quote can therefore be reworded as
A brief remark hear unpredicatably or sigh are just as much evidence as a speech or a murder. — E.M. Forster
With a quick Google (
a chance word or a sigh e.m. forster) I found this page with a fuller version of the quote:
“Character,” says Aristotle, gives us qualities, but it is in actions—what we do—that we are happy or the reverse.” We have already decided that Aristotle is wrong and now we must face the consequences of disagreeing with him. “All human happiness and misery,” says Aristotle, “take the form of action.” We know better. We believe that happiness and misery exist in the secret life, which each of us leads privately and to which (in his characters) the novelist has access. And by the secret life we mean the life for which there is no external evidence, not, as is vulgarly supposed, that which is revealed by a chance word or sigh. A chance word or sigh are just as much evidence as a speech or a murder: the life they reveal ceases to be secret and enters the realm of action.
They cite his book Aspects of the Novel. Now with a fuller context we can understand the quote better. Here "action" means actions taken in the world, which others can see. This contrasts the "secret life" which is internal to a character. Random words or body language ("a sigh") can move this internal life into the external realm. They're "just as much evidence" as a more obvious external action, such as a "murder", when it comes to developing character. So when demonstrating to a reader what a character is like (moving the secret, internal life into the external world), random words and body language are as effective as more obvious external actions.