To unlock you must visit the Sol system but you need a permit. To obtain this permit you much reach the rank of Petty Officer in the Federal Navy. To obtain this rank you must do missions for federation aligned minor factions at stations in federation controlled space. The more successful missions you do at multiple system stations, the faster your rank will increase. You can tell if the missions are Federation aligned in the missions tab at stations because they will have the Federation symbol with the party giving the mission. You will be able to check your progress with the Federation in the systems tab in your ship. To progress to the next rank you will be given missions that usually say something like "Do -X- for the Federal Navy. Get to the rank of Petty Officer (about 4 levels of rank) and you will get the permit. Then fly to Sol.
Note: sometimes after obtaining the rank of Petty Officer it will not show that you got the permit. If that happens, log out then log back in and it should find it in your systems tab