Basically, it's a very long and slightly peculiar way of saying yes (ja).
It's not that strange if you know a bit about Swedish linguistics, as regular readers of this column will do. The Swedes are big fans of doubling up words and saying thinks like hej hej and tack tack. This kind of repetition can be used for emphasis or a change in tone – in these two examples, it makes the statements slightly more enthusiastic.
Answering questions with ja ja or even ja ja ja is not uncommon in Swedish, and means 'yes yes yes!' or 'yes, absolutely'. But because ja ends in a vowel, it comes more naturally to repeat the word than 'yes' does – your mouth doesn't need to move so much.
To make your 'yes yes' even more decisive, you can add on mensan. This is the interesting bit of the word. You'll also hear jajamensan shortened to just jajamen, but it doesn't come from men meaning 'but' (in the way that in French, mais oui! (but yes!) is used for emphasis, for example).
The mensan has evolved from an Old Swedish expression, a mina san (literally 'in my truth'), which became på min sann (by my truth) in Modern Swedish – and by modern, this was a phrase used in the 1500s. Over time, with repeated use the expression was eroded to become minsann, which since the 1600s has been used as an intensifier to stress the truth of something. For example, det är minsann inget att skratta åt (that is certainly nothing to laugh at), or Jag är minsann inte förvånad (I'm not surprised, to say the least).
So when put together, jajamensan can be literally translated as something like 'yes, yes, truly!' It's a way of agreeing strongly and emphatically: other alternatives would be ja, precis! (yes, exactly!) or absolut! (absolutely). But neither of those are quite so fun to say.
A word of caution. Jajamensan is very colloquial and informal Swedish, but it's not the most modern or cool language – you're more likely to hear a middle-aged parent say it than their teenage children. An English equivalent might be 'yes, indeedy!'
And for those of you wondering if there's an equivalent for the word 'no'? Jajamensan, there is, and it's nänämensan.
Vi var inte säkra, men sen googlade vi och jajamensan, det stämmer
We weren't sure but then we googled it and yes indeed, it's right
Är du redo? Jajamensan!
Are you ready? I sure am!