In the first example above about Puma, I wanted to quote something she says. However, because echo statements enclosed by double quotation marks are interpreted by PHP to run as PHP, to include additional double quotation marks within the statment makes it confused and creates an error. This is why I used the backslash so that the PHP would ignore interpreting the double quotation marks that quote Puma's statement as something to run, and instead just have them appear on the page with the rest of the text.
In the second example about Pickle I used the escape slash to include dollar signs. Again, similarly to example 1, this tells PHP to ignore running the dollar signs as if they are to reference a variable to be passed through the string, and instead just echo them out with the rest of the text.
In the third example refferring once more to Puma I use an escape slash within an echo statement using single quotes to ignore the single quote used within the string so that instead of throwing an error, the single quote is displayed properlly with the rest of the text.