meaning - What does the phrase "prospects were beginning to chirp back" mean in this sentence?

For reasons unknown, I am unable to access the reference link provided by you.

Nonetheless, I can infer that the article is about how to write e-mails to customers.

You seem to have problems in interpreting the expression - "to chirp back".

As you've pointed out, the ODO defines chirp as

(Of a person) say something in a lively and cheerful way

In this context, it simply refers to a response. Whether it was cheerful or lively, as readers, we are not concerned.

Taking the sentence in question,

But I slowly started to notice a change. Clients were increasingly respondent to my emails. Even prospects were beginning to chirp back more times than not.

The author is probably giving out e-mail etiquette/ writing tips when sending mails to customers. By following those, the author noticed that customers, who generally don't respond to e-mails soliciting their feedback or product promotions (e-mail of any kind, for that fact!), start to respond to them and even "prospects"(potential customers of a business) were starting to reply to messages and perhaps reacting to promotional campaigns("chirping back"). More times than not means "often" or "frequently".