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The DHCP client initially locally broadcasts for the DHCP server. This broadcast has the MAC of the client.
This is also why the DHCP server either needs to be on the same network or you need forward this broadcast to the DHCP server if it's non-local. The latter is normally accomplished with an IP helper.
The client's broadcast contains the client's MAC, so this is how the DHCP can L2 unicast back to the DHCP offering it an available IP address out of a pool of IP addresses it maintains. If an IP helper was used, the DHCP server also has the helper's IP address so it sends back to the helper which will forward the response back to the client.
