GPG signatures are a proof that distributed files have been signed by the owner of the signing key. For example, if this website was compromised and the original Electrum files had been replaced, signature verification would fail, because the attacker would not be able to create valid signatures. (Note that an attacker would be able to create valid hashes, this is why we do not publish hashes of our binaries here, it does not bring any security).

In order to be able to verify GPG signatures, you need to import the public key of the signer. Electrum binaries are signed with ThomasV's public key. On Linux, you can import that key using the following command: gpg --import ThomasV.asc. When you import a key, you should check its fingerprint using independent sources, such as here, or use the Web of Trust.