I'll try to be as objective as possible in the answers:
What is a TXT entry for?
A TXT record is a resource record type in DNS that contains text information from sources outside your domain used to associate arbitrary text with a host such as readable information about a server, network, and so on. You can use TXT records to prevent phishing and spamming.
How do I read the TXT configuration for a given domain?
As Jose Maia said, to read the configuration, it is done in the same way as any other DNS lookup, i.e. NSLookup.exe -q = TXT example.com or dig -t txt example.com.
Who cares if I have a TXT set up in my domain?
The domain host maintains the DNS records, which direct Internet traffic to its domain name. When you add a record check TXT DNS records from your domain host. if you do some verification (for example by Google to use services like G Suite), your domain ownership is confirmed.
What are the advantages of knowing this in day-to-day programming?
The main focus is to have an overview of whether you have a TXT record confirms that you are the owner of your domain, for example when Google or Office365, etc., see the record, the ownership of the domain is confirmed.
How many TXT records can I have per domain?
A domain can have multiple TXT records associated with it, as long as the implementation of the DNS server supports this. However, you can not have more than one TXT record for the SPF (Sender Policy Framework) of a domain. If the domain has more than one SPF record, you will receive email errors as well as spam rating and delivery issues.
Note: The verification log does not affect your website or email.