Where can I read the messages people send through my contact form?

They are saved in your admin. This needs the Form Builder plugin.

• Go to Settings → Form Submissions → Contact Form Submissions and pick the form from the Form dropdown. The number of responses is shown next to each form name. Each response appears as a row with its date and answers.
• To download them, open the form's settings (the gear icon in Contact Forms), go to the Submissions tab and click Export CSV.

To get an email for each new message, open the form's Email and Actions tab, turn on Admin Email and enter the address under Send admin notifications to. These emails are only sent when the form has a working reCAPTCHA. Without it, the email log shows Not sent — no reCAPTCHA, but the submission is still saved.