You're on an airplane that's about to take off. There's a video explaining what to do in different emergency situations. The video is showing how to use an oxygen mask. The speaker on the video says this.
Always secure your own mask before assisting others.
In this case, "my own" means something that you don't share with other people. So your own apartment would be one that you live in by yourself.
The word "assist" means "help" but is more formal. Use it with customers and other people that you need to speak to respectfully:
How can I assist you?
Can you assist this gentleman with his question?
To "secure" something means to keep something safe by making sure that it doesn't move, fall, slip, etc.
You can "secure" something by tying it, strapping it in, locking it, and so on. For example:
Make sure to secure the straps on your harness.
After you're finished, replace the cover and secure the bottom screws.
You can also use "secure" to mean "protect" in a wider range of situations:
They feel that the country should be making more of an effort to secure its southern border.
"Others" is a formal way to say "other people":
Please be respectful of others.
Kids who go to preschool learn to play well with others.
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