Several years ago, I participated in a learning seminar in tropical algebraic geometry and collected several helpful survey articles. (This was before Maclagan and Sturmfels' book was written, which I suspect is excellent.)
Anyway, here were some of the most helpful intro points for me:
First Steps in Tropical Geometry,
Tropical Algebraic Geometry,
Introduction to Tropical Geometry,
The Tropical Grassmannian,
The Number of Tropical Plane Curves Through Points in General Position.
Sturmfels, Speyer, and Gathmann all write very well, and Gathmann especially devotes considerable space to giving motivation for the field. Mikhalkin, of course, was the one who pioneered the idea of attacking challenging classical problems (such as counting the number of plane curves of genus $g$ and degree $d$ passing through $3d + g - 1$ points, which had just been solved by Capraso-Harris in the late 90s) using the tropical semifield.