In case it wasn't obvious, the stuff that goes above and below a "big operator" are treated as subscripts and superscripts in LaTeX. You can insert these just by inserting the "big operator" along with the normal LyX functions for superscripts and subscripts immediately afterwards.
Not just in LyX, but in LaTeX generally, subscripts and superscripts are written next to, not above/below big operators in inline math rather than display math. If you want to get the superscripts and subscripts above/below in inline math, you need to use the \displaystyle command. LyX has a button for that under "Styles". You then have to click inside the little box it creates and insert the big operator in there.
Here's a little animation showing how it works. Let's see if this works on the forum (scroll to the bottom to see what buttons I'm pressing):

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