In more detail
A sending server looks up the MX records of the recipient's domain and tries the server with the lowest priority number first, falling back to the others if it is down; servers with equal numbers share the load. MX records must point to a hostname, not directly to an IP address, and that hostname must not be a CNAME. If a domain has no MX record at all, senders fall back to its A/AAAA address; a domain that never receives mail can say so with a "null MX", MX 0 . (RFC 7505).
Why it matters
When switching email providers, a wrong or forgotten MX record means mail keeps going to the old provider or bounces. Lowering the TTL a day before the switch shortens the overlap. MX records only route incoming mail; whether outgoing mail is trusted depends on SPF, DKIM and DMARC.
Example
example.com. 3600 IN MX 10 mx1.example.com.
example.com. 3600 IN MX 20 mx2.example.com.
Senders deliver to mx1.example.com (priority 10) and use mx2.example.com (priority 20) only when the first cannot be reached. See How DNS Works.