There are two ways to ship color information on a mesh:
A): Per vertex color - each vertex of a mesh owns a certain RGB color. The color of a triangle is the result of blending the 3 colors of its vertices. In this case there's only a .obj file.
B): Texture - there's an additional image next to the .obj file. Each triangle of the mesh points to a certain region of that image (somehow like a snippet of a wallpaper). In this case there's a bundle of three files: the geometry (.obj), a material definition file (.mtl) which links the obj to the third file which is some image format (.jpg, png or ...).
In case of B: MM automatically loads the texture shader (a world sphere shader from SHADERS) as soon as you import the file in MM. If MM fails to load a texture there's something wrong with the link chain of the three files (in most cases the whole bundle needs to be in the very same directory).
In case of A: To show per vertex color you need to switch the default rendering mode:
Press Space to open the Hotbox. In its Color section click on the sphere with the blue gradient
or
Go to Preferences via menu and on its General tab set MeshColorMode to VertexColor
