That depends on what client software you’re using. There are over 50 options, and none of them are actually called “Let’s Encrypt” now, as that name is used only for the certificate authority service.
For most people in most situations the easiest option is simply to get a new certificate on the new server and completely ignore the existence of the old certificate. There is no restriction on the simultaneous existence and validity of overlapping or identical certificates, except for Let’s Encrypt’s own rate limits
However, if you only have one site and the server migration is a one-time or infrequent event, the rate limits shouldn’t affect you at all in this circumstance.
If you do want to keep using your existing certificate or setup, you’ll have to backup certain directories and make sure that they’re installed on the new server in the same way that they were on the old server. For example, with Certbot, you can back up and migrate the
/etc/letsencrypt directory (including symbolic link structure). Then Certbot would see the old certificates and configuration and continue using them, assuming that the configuration of the server is otherwise similar enough that whatever authentication method was used to prove control of the domain also works in the same way on the new server. But again, most people in most circumstances would probably find it more straightforward to start from scratch.