What does ‘May that not be the takeaway of this speech’ mean in Salah Palin’s mocking Michelle Obama’s recommendation of breastfeeding?

Solution 1:

It has nothing to do with food. Takeaway in the sense of food is the British version of the American takeout, which is food ordered from a restaurant but picked up and taken away to eat at home.

The takeaway of something is something like the gist:

gist — the substance or essence of a speech or text : she noted the gist of each message.

But you've probably gathered as much already. What I think is confusing to you is this construction:

... and may that not be the takeaway ...

Here may is used as a wishful statement, expressing the hope that a thing may happen. There is probably no better way to illustrate this than with the traditional Irish blessing:

May the road rise up to meet you, may the wind be ever at your back. May the sun shine warm upon your face and the rain fall softly on your fields. And until we meet again, may God hold you in the hollow of his hand.

The speaker here is expressing the wish or hope that all these good things will happen to the recipient of the blessing. If you were to write it another way, it would be something like

I pray it may happen that the road will rise up to meet you ... etc.

In a different light, prisoners condemned to be executed used to be read the statement of their sentence (i.e. punishment by being put to death) and at the end of that statement the official would say, "And may God have mercy on your soul." This was not a blessing, like the Irish one above, but something more like a pro forma dismissal.

In view of all this, Sarah Palin was telling people she hopes that the laugh she got from the crowd would not be all that was remembered from what she said. In a sense, she wanted to get the laugh from her statement but not really take responsibility for it.

Solution 2:

In this context, takeaway means the message or lesson that is taken away from an experience; that which is remembered from it. It may or may not be the "essence", but it's the part that stays with you. Or, to repeat the idiom, that you take away with you.