To add to that great advice...
Those Euphorbia fruits should turn yellow or brown when they are ripe. At a time of their choosing, they will explode. The fruit splits into 3 or more parts and the seeds go flying. Sometimes you can find them later, usually they are lost. Then you end up with volunteers growing in nearby pots a couple years later. This behavior resembles Dorstenias, which use a very similar trick for seed dispersal.
So you have two options: (1) collect the fruit when it is almost ripe, but before it splits open; or (2) trap the seeds somehow, with a big plastic box or plastic bags, or a drop of school glue on the fruit so it doesn't break open when it ripens. Your creativity is worth cultivating with this second option. I believe Kaktus documented some of his experiments in this thread, so you can refer to those and see what ended up working well for him.
In order to harvest the fruit intact, you need to be watchful for a change in color. The green will start to go yellow and/or brown. Once the fruit is close to full size, that color change may take 2-3 weeks to occur, or much less.
Anyway, you might have 24 hours or less from the color change until the fruit pops. So you have to check regularly. Like every day. It doesn't have to be all all the way to a full color change, either. Once a fruit is partway there, you can often collect it and get full ripening on the shelf, at least some of the time.
In all cases when removing unripe fruit, use a sharp tool (like nail scissors) and be prepared for a bit of a gush immediately after you snip. Avoid that sap and clean your tool right away afterwards.
Put the harvested intact fruit in a paper envelope in a place with some warmth and good air flow. Over the course of the next few days it will usually do its thing and you will find the seeds inside the envelope. If it's hesitant or slow, you can sometimes coax things into action by putting the envelope in a warm, dry place (like in the sun on a warm day). It makes an audible pop when it explodes, so I usually leave them on my desk where I work, and then I get to enjoy a sharp popping sound when the magic takes place.
Those look like pretty big fruit (for a Euphorbia anyway) and that usually means the seeds are also big. And that is a good thing because it means they will get started earlier and faster, all things equal.
