There are two cases here:
Standalone Scripts
Install them under bin using bin.install. You can optionally rename them, e.g. to strip the extension:
class MyFormula < Formula
  # ...
  def install
    # move 'myscript.sh' under #{prefix}/bin/
    bin.install "myscript.sh"
    # OR move 'myscript.sh' to #{prefix}/bin/mybettername
    bin.install "myscript.sh" => "mybettername"
    # OR move *.sh under bin/
    bin.install Dir["*.sh"]
  end
end
Scripts with Support Files
This case is tricky because you need to get all the paths right. The simplest way is to install everything under #{libexec}/ then write exec scripts under #{bin}/. That’s a very common pattern in Homebrew formulae.
class MyFormula < Formula
  # ...
  def install
    # Move everything under #{libexec}/
    libexec.install Dir["*"]
    # Then write executables under #{bin}/
    bin.write_exec_script (libexec/"myscript.sh")
  end
end
Given a tarball (or a git repo) that contains the following content:
- script.sh
- supportfile.txt
The above formula will create the following hierarchy:
#{prefix}/
  libexec/
    script.sh
    supportfile.txt
  bin/
    script.sh
Homebrew creates that #{prefix}/bin/script.sh with the following content:
#!/bin/bash
exec "#{libexec}/script.sh" "$@"
This means that your script can expect to have a support file in its own directory while not polluting bin/ and not making any assumption regarding the install path (e.g. you don’t need to use things like ../libexec/supportfile.txt in your script).
See this answer of mine for an example with a Ruby script and that one for an example with manpages.
Note Homebrew also have other helpers to e.g. not only write an exec script but also set environment variables or execute a .jar.