The quest can be unlocked only if your previous actions led to the medium ending in Tesham Mutna main quest. In this ending Syanna is killed by Dettlaff and it happens if while in the fairy land (Beyond Hill and Dale quest) Geralt didn't obtain the magic ribbon for Syanna. The protagonist will be punished by going to Prison in Toussaint (M5,13).
Important - If Syanna is still alive, then instead of Burlap is the New Stripe quest another one will activate - Pump and Strange Circumstance about meeting Anna Henrietta in the palace.
Activities on prison's main courtyard
You will start the visit in the prison by speaking with Jacques who will tell you to go to the laundry. Before you do it, check the main courtyard of the prison. First find the loose part of the wall shown on the picture. It is a stash with Notes found in the Toussaint prison hidden inside. Reading it will unlock a short additional quest The Perks of Being a Jailbird. Complete it now.
Remain on the courtyard. Geralt must find Small hammer and chisel for carving stone. These items are located in the wagon shown on the picture 1. After taking it you must place it in the stash shown on picture 2. After completing these actions Geralt must wait seven days, so use medication. Go back to the stash after full seven days and take Prison Poster from it. After leaving the prison you will be able to hang it in Geralt's mansion.
In addition to solving the quest described above you can also speak with the prisoner who plays gwent. If you win with him (he uses Skellige deck) you will obtain bread.
Escaping the prison
Go to the laundry after completing all activities on the courtyard. There Geralt will have to complete some trivial actions - he must pick up the soap and the dirty clothes, move them to the container, take the stick and start washing. Hard work will be stopped when a group of prisoners will appear in the laundry. You can use fully developed Delusion ability on them or immediately start fighting them. Use the stick instead of the regular silver sword.
After another cutscene you will have to follow one of the guards. It is a very linear part of the quest. The guard will lead Geralt to the place where he will meet with Damien and with Dandelion. Dandelion will tell you that he managed to secure Geralt being released from prison. Follow the guard to the exit gate and check the chest on your way in order to take all of your belongings.
Regis will be waiting for Geralt before the prison. During the conversation the vampire will suggest you to identify the person that was supposed to be the fifth victim of the Beast of Beauclair. You can agree or decide not to do it. If you decided to do it, then you must meet with the shoeshiner. This option is thoroughly described in the walkthrough for Pump and Strange Circumstance quest in the Speak with bootblack section.
The final part of the epilogue is same no matter whether you obtained information about the fifth victim and shared it with Anna Henrietta or not. Geralt will meet with Regis on Mere Lachaise Cemetery. You will have a short talk with friend and will have to complete a simple task of collecting two Mandrake roots. As Regis suggests, wear mask and gloves before doing it. If you skip this step, then Geralt will be seriously poisoned while trying to take the roots. Use Witcher Senses to find the plants. In the meantime Regis will disappear, but simply follow the footprints to find him again. Geralt's friend will take care of the vampires that attacked him.
At the end of the quest you can answer Regis' questions about what will you do now. No matter the answer you will unlock a very short quest Be it Ever so Humble.