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".