Old dial-up modems make weird sounds when connecting to the internet because they are performing a "handshake" process to test the telephone line quality. During this process, the two modems send precisely specified sounds and listen to what is heard on the other end. This allows them to negotiate settings such as speed and compression methods. The noises are also used to transfer data in an analog format, which was common during the early home PC era of the 1970s and '80s. 

So basically, old dial-up modems make weird sounds when connecting to the internet because they are testing the telephone line quality and exchanging signals with each other in a process known as handshaking. The modem sends precisely specified sounds and the other listens to see what it actually hears on the other end. This noise was the analog world's way of transferring data.
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