The word takeaway has two meanings, the first one is the food that delivered to you by the restaurant in which you ordered. For example order some chinese takeaway. The second one is the insights, knowledge or understanding you get from someone's writings or speeches. For example, a blogger should give the readers takeaway regularly, or visitors will leave the website.
The word takeaway also means a key point or a key fact. When you asking what's the takeaway, is the same as asking what's the point. Its the same as lessons, important things, points or messages delivered by speeches, representations, articles and books. The takeaway is fancy way to express those ideas.
The dictionary defines the word as
The lesson or principle that one learns from a story or event. Used with the.
A key fact, point, or idea to be remembered, typically one emerging from a discussion or meeting.