In plain words
Your mail server signs each message; the receiving server checks the signature against a key published in your domain’s records.
Why it matters to a firm
Without DKIM a DMARC policy has less to go on, and mail sent through a marketing tool or booking system in your name can fail or be forged.
How anyone can check it
Send yourself an email, open it and view the original message or headers. Look for dkim=pass and the domain after d=.