How does bitsquare sell/buy fiat currencies when it is decentralized? I noticed that on bitsquare you can sell your BTC for fiat such as usd or euro. But my question is how is that decentralized when you want to cash out with USD or EUR? Who is the middle man that interacts with the bank to do your fiat cash outs? 

And second of all how do you get fiat into the system in bitsquare in the first place? Who issues these fiat tokens in usd or eur?

There has to be "someone" that issues these fiat tokens as how do users get fiat tokens in the first place? If they deposit fiat, you need a 3rd party to verify those deposits are legit. Who is this entity?
In this situation you are the third party who verifies those deposits are legit. All banking transactions take place directly between the buyer and seller. Bitcoin goes into an on-chain 2of3 multi-signature address (in the case of bitsquare and cancoin) or a custodial wallet (in the case of localbitcoins). Once the bitcoin is in escrow the seller gives their bank payment details to the buyer and the buyer initiates a bank transfer directly to the sellers bank account. At this point the seller will release the bitcoin from escrow and send it to the buyer.