From man bash:
Enclosing characters in double quotes preserves the literal
value of all characters within the quotes, with the exception of $,
`, , and, when history expansion is enabled, !.
In other words, " ... " is not sufficient to protect the unbalanced backtick on line 4 of your text; the shell is interpreting it as the start of a command substitution.
OTOH you can't use single quotes, because your text contains single quotes.
I'd suggest avoiding the issue of quoting altogether by using a here document. You should also use a shebang to make sure your file is interpreted by the intended shell. So:
#!/bin/sh
cat <<'TXT'
_ ____ __ _
_ __ () / ___|_ __ __ _ / | |
| '_ | \ / / | | '/ ` | || |
| | | | |> <| |___| | | (| | _| |
|| ||//_\____|_| \,|| __|
TXT
then
$ ./mat.txt
_ ____ __ _
_ __ (_)_ __/ ___|_ __ __ _ / _| |_
| '_ \| \ \/ / | | '__/ _` | |_| __|
| | | | |> <| |___| | | (_| | _| |_
|_| |_|_/_/\_\\____|_| \__,_|_| \__|