what's the purpose of ssh-agent?

The SSH agent handles signing of authentication data for you. When authenticating to a server, you are required to sign some data using your private key, to prove that you are, well, you.

As a security measure, most people sensibly protect their private keys with a passphrase, so any authentication attempt would require you to enter this passphrase. This can be undesirable, so the ssh-agent caches the key for you and you only need to enter the password once, when the agent wants to decrypt it (and often not even that, as the ssh-agent can be integrated with pam, which many distros do).

The SSH agent never hands these keys to client programs, but merely presents a socket over which clients can send it data and over which it responds with signed data. A side benefit of this is that you can use your private key even with programs you don't fully trust.

Another benefit of the SSH agent is that it can be forwarded over SSH. So when you ssh to host A, while forwarding your agent, you can then ssh from A to another host B without needing your key present (not even in encrypted form) on host A.

The benefit to ssh-agent is that you only need to enter your passphrase once. If your private RSA key is not encrypted with a passphrase, then ssh-agent is not necessary. The ssh command would be an example of a client.

If you are routinely sshing into a variety of different machines, each with their own key and passphrase, then running ssh-agent allows you to enter the passphrase for each key once¹ at the start of your session and then you can authenticate to each machine as many times as you like without having to re-enter your passphrase.

A further benefit is that, as per the man page, the agent never sends a private key over its request channel; so if you are hopping between different boxes, your private keys are protected.

¹You can set the life time that the keys are held in the agent.