For an intro to all things dapps, see our user guide here.
For sites to see your wallet address, they must ask you for access. All they're gaining in these situations is the ability to see your address, and, by extension (since everything is viewable on the blockchain), the funds you hold. This does not mean they can access and move your tokens around — for that, you'll need to sign a token approval.
You will only be required to connect to a dapp once. After that, you'll be able to see it in your 'Connected sites'. For more detail on viewing connected sites and for how to disconnect from them, see here.
Some legacy sites may not request access properly. If you go to a dapp and it does not see your wallet, you'll need to add it manually. See our article for how to do this.
For more information on dapps, see here.