The My Places page is where you can see the information about your own Places. Here, you will see a list of all the places that you own in your Showcase. These places are the places that other people can see as well, on your Profile Page.
On each Place you will see a bunch of different things. Lets run through them.
The giant buttons
I'm sure you've noticed those already. These buttons are to select what you want to do. Sometimes some buttons are shown, sometimes they aren't.
Play Button
This giant button is what you press when you want to play online right now. It will start Roblox Studio and load you into the first empty Server that is available. If no spaces are available, it will start a new Server for you and your friends to play on automatically.
Build Button
This slightly smaller button is the one you press to go to that place by yourself. It starts Roblox Studio and lets you play around in the places. This button will only show up if you own the place, or if the owner has the place set to Un-copylocked.
Edit Button
The Edit button only shows up on your places, or Un-copylocked places, and only if you're looking at the website through Roblox Studio. This will open the Place in Edit Mode, where you can really get some building and scripting done.
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This is an example Places list, from Doom's places. Click on something to go to that section on this page.