Apple - How can I get the exact URLs/requests that a program is trying to connect to?

The most full featured way to peek into web traffic is to install and configure your system to use a web proxy.

The mitmproxy proxy (and mitmdump tool) is mature, reliable and well documented so I would start there unless you have another that meets your specific needs.

Also, widely used, superbly documented and well regarded is Charles Proxy

Sounds like you need a packet sniffer.

Wireshark will log and do full disassembly on all traffic going through your network interface.

In other words, it will not just show you the full URLs of each request, but it will let you look inside all the packets, so you can see the headers and content too (except in SSL connections, as they are encrypted).

It works on just about every platform out there, including Macs.

Whilst it won't give you the full URL, if you're a fan of the command line, then iftop also comes in quite handy and shows you both incoming and outgoing hosts, port numbers (names for well known ports) and cumulative totals for each - as well as live updating "graphs" on the command line :-)