GitHub Pages doesn't offer/support anything like this, since it's just a static website hosting. It serves HTML files and assets, but doesn't run server side scripts.
A few months ago I run into the same problem and did some research. As others mentioned, an external form handler service is needed for this task. Fortunately, there are tons of them:
I compared their features and limits based on their website, then I chose, tried out Basin and finally I stuck with it. It offers spam filter, captcha verification, Zapier integration; hides your email address in your form and it can send an email receipt to the one who submitted the form - all of this for free. It works like a charm on my website (hosted on GitHub Pages) behind my contact form.
EDIT: It offered unlimited submissions when I originally wrote this answer, but now a paid subscription is required for unlimited submissions and a redirect URL. Free tier now includes 100 submissions/month.
It's very easy to set up, the administration interface shows HTML snippets to copy-paste. Your form will look something like this:
<form accept-charset="UTF-8" action="https://usebasin.com/f/123456xabcdef" enctype="multipart/form-data" method="POST">
<input type="email" id="email" name="email" required>
<textarea rows="4" cols="50" name="comment" required></textarea>
<button type="submit">Submit</button>
</form>
Some time ago I wrote a blog post about testing Basin with more details.