The exchange transaction between my Exodus wallets BTC to ETH.
That wasn't the question. He wasn't asking for the trading pair but the name of the exchange you used. The service provider. Was it ShapeShift or some other instant exchange available from within the Exodus wallet?
The question is how did they withdraw the funds from my wallet? Did they logged in into my account?
I don't think so. You tried to swap the coins. That means a transaction was made from your wallet to that of the instant exchange that was used. That's where the swap happens. When it's done, the exchange sends you your new coins to the wallet address you provided. This all happens in the background and you don't see it. Since your assets were flagged for whatever reason (my guess is because the amount is quite big), the instant exchange froze your crypto and is now asking who you are and how you got it.