The tipical, and also most useful, example use case for FHE is running computational tasks on some cloud service without having to trust it. And yes, it would provide protection against Meltdown and Spectre (if performed on the hardware running the computation), as the attacker would be able to only extract encrypted data.
The data has to be decrypted at some point in order to display it... unless we're envisioning FHME hardware in the monitor as well - honestly I think we're well across the threshold into fantasy already though.
Of course the data has to be decrypted, but in this case you would decrypt it on your client machine, so that you don't need to trust the cloud provider or other third parties using VMs on the same server (side channel attacks can sometimes be exploited from another VM running on the same hardware, although this is rarely considered as part of one's threat model).
