Transports are fast, cheap and durable. I use them to move my fleet, attack and scout.
Moving ships like Missile Frigates and Plasma Siege Starships can be painfully slow. Especially if the distance is long and/or you need to go to defend one of your planets from an incoming wave and/or destroy a Mining Golem.
They can also be used when attacking. For example.. I spot a Super Hybrid 4 hops away from one of my planets. I can just transport my fleet to the planet where the Super Hybrid is without my ships taking any damage.
Transports also help you to scout planets you otherwise couldn't scout. Put a cap of MarkII scouts (20) into a Transport and send the Transport where you want. When the Transport is destroyed pause the game and move your scouts. That way you can scout few extra hops. Very useful. In my
current game I've almost scouted the whole map.
EDIT: Oh and they help you destroy Data Centers! Transport your fleet on a planet with a Data Center. Destroy the Data Center. Gtfo