1. why do we have emotions?
so-called ‘negative’ emotions – like anger, panic & anxiety – prepare the body-mind to deal with specific short term threats by narrowing down our thought-action choices
so-called ‘positive’ emotions – like joy, contentment & curiosity – broaden the thought-action repertoire by promoting bonding, play, exploration & integrative, creative thinking to build resources for future survival
adrenaline
& dopamine
Ashby, F. G., A. M. Isen, et al. A neuropsychological theory of positive affect and its influence on cognition. Psychol Rev 1999; 106:

2. ... improving problem solving
positive emotions have been shown to help people think & problem solve with more speed, creativity, flexibility, curiosity, efficiency and organization
Isen A M, Johnson M M et al. The influence of positive affect on the unusualness of word associations. J Pers Soc Psychol 1985; 48:
Isen A M, Daubman K A et al. Positive affect facilitates creative problem solving J Pers Soc Psychol 1987; 52:
Isen A M, Rosenzweig, A S et al. The influence of positive affect on clinical problem solving. Med Decis Making 1991; 11: 221-7