The answer below was written under the assumption that you could get MathML markup from Math Input Panel. This turned to be false.
The Math Input Panel doesn't offer fallback text representations of the XML markup on the clipboard. Marko Panic, the program manager for the development of this tool confirmed to me that this was a design decision as they didn't want the end user to be faced with raw XML.
What a wasted opportunity. The Input Panel generates MathML markup for your formulas, but it won't give it to the user. It could be that seeing an XML tag was found to cause brain damage in Microsoft users, I'm not sure. The result is that the Input Panel works only with expensive products like MS Word and MathType, which are designed to receive the data in the way that the input panel produces.
If you don't mind spending $97 on MathType, then that may be a solution: MathType claims to produce LaTeX in a way that is "MathSE-compatible". I'd rather not link to their site.
In case anyone is interested in what Math Input Panel looks like (I never saw it until today), here is a screenshot. The insert button is the only thing that outputs the markup, and it bypasses the user, sending data directly into the application that currently has focus.
Old answer follows.
Math Input Panel outputs your equations in MathML format. MS Word, its OpenOffice analog, and some other software can render MathML. Browsers, for the most part, cannot do it natively, but with the MathJaX script they do. The main problem is that SE is not configured to allow MathML tags as user input.
Suggested workaround: try to convert MathML to LaTeX and then paste LaTeX code here. Unless the Input Panel can produce LaTeX directly...
An additional complication to the above is that the MarkDown parser used by SE kills all XML tags except for a very limited set. MathML looks like this:
but after the MarkDown parser does its job, what remains is
Yet there is hope
The parser knows to spare the content between dollar signs. E.g., in
$<math> </math>$ MathJaX gets to see the tags. MathJaX is perfectly capable of rendering MathML, but unfortunately it concludes from the dollar signs that the content is TeX, and we get $<math> </math>$. This suggests two plausible solutions:
MathJaX should understand that
$<math> ... </math>$ is MathML pretending to be TeX to avoid MarkDown.
MarkDown parser should treat
<math> ... </math> the same way it treats
$ ... $ now, i.e., ignore it.
The second solution is cleaner: you'd be able to paste MathML without surrounding it with extraneous dollar signs. It is also better future-proof, if in the future browsers begin to support MathML directly and scripts like MathJaX will no longer be needed.
I'd like to see the second solution implemented here. It would take some work, but I think it's doable.