EdDSA is preferred over ECDSA/DSA for SSH or any other secure protocol. ECDSA relies on a random number nonce which if found could allow the private key to be derived.
What Makes Them Different? RSA: Integer Factorization DSA: Discrete Logarithm Problem & Modular Exponentiation ECDSA & EdDSA: Elliptic Curve Discrete Logarithm Problem
The computational complexity of the discrete log problem allows both classes of algorithms to achieve the same level of security as RSA with significantly smaller keys.
So effectively ECDSA/EdDSA achieve the same thing as RSA but with more efficient key generation and smaller keys. They are not inherently more secure than RSA.
Functionally, where RSA and DSA require key lengths of 3072 bits to provide 128 bits of security, ECDSA can accomplish the same with only 256-bit keys. However, ECDSA relies on the same level of randomness as DSA, so the only gain is speed and length, not security.
Bit security measures the number of trials required to brute-force a key. 128 bit security means 2128 trials to break.
Taking a step back, the use of elliptic curves does not automatically guarantee some level of security. Not all curves are the same. Only a few curves have made it past rigorous testing. Luckily, the PKI industry has slowly come to adopt Curve25519 in particular for EdDSA. Put together that makes the public-key signature algorithm, Ed25519.
Curve 25519 was chosen because it was believed that the NSA had potentially implemented a backdoor into the random number generator used by ECDSA. See https://en.wikipedia.org/wiki/Curve25519
Source https://gravitational.com/blog/comparing-ssh-keys/
Read up on how the SSH handshake works in a related blog post https://gravitational.com/blog/ssh-handshake-explained/ and https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange .