How to unlock: Automatically after you meet with Virgil in Rocky Cave (M8,1) during quest The Glowing See.
This quest does not appear in your journal, but it requires you to perform a number of actions and, as a result, it is a good idea to take interest in it, of your own. During the main storyline, you will unlock a quest - The Glowing See, which requires you to find scientist Virgil. You can find him in Rocky Cave and it is required that you meet him, to get information on how to get into the Institute. During the conversation, Virgil asks you to find a cure that would help him cure the unwanted body mutations.
You can try and find cure for Virgil no earlier than after you have progressed into the storyline to the point, where you receive quest Institutionalized, in which you meet Father in the Institute. During the meeting, you need to state that you want to work for the Institute, because you can then move freely around the location.
At any moment now, you can find, in the Institute, door to Bioscience department. Here, find the blocked door to the FEV Lab, hack into the terminal and enter. Take interest in the main room in the FEV Lab, where there are mutants in large pods filled with green substance. The Experimental Serum that you are looking for is in the small container on the desk - it has been shown in the above screenshots.
Note - You can access the FEV Lab after you start a side quest for the Institute - A House Divided. Newton Oberly, opens an alternative route for you that does not require hacking into the computer. However, in this situation, you need to keep in mind that you will have to destroy all robots and turrets along your path.
All you need to do now is return to the Rocky Cave and give Virgil the serum. Act accordingly with Virgil's request and wait for around a week of the game's time (you can use the bed several times) and meet with him again. Taking the cure ends in a success and Virgil returns to his previous form. From now on, you will be able to use Virgil's lab and take the items in it.
Rewards for completing the quest: full access to Virgil's lab
Note - Also, there is an alternative ending to the quest. During your last conversation with Virgil, you can use Charisma and lie to him that you were unable to find the cure in the Institute.