The Twilio PHP Helper Library is a tool used for sending SMS messages in PHP applications. To install it without using Composer, one needs to create a new directory, download and decompress the library, test its installation by sending an SMS, purchase a Twilio phone number, write PHP code to send an SMS, and run the script. The process involves logging into the Twilio console, finding a phone number, purchasing or obtaining a phone number, writing PHP code with placeholders for Account SID and Auth Token, substituting these values, testing the script, and running it in a console. This method allows developers to use the library without relying on Composer.