Those lines are called a "Graticule" the particular shape you get depends on the particular projection you use, which depends on the extent of your map (Where it is and how big it is), and the particular properties you are trying to preserve (Angles, distances through a point, areas, etc)
That map is using a Conic projection. I think it's probably Lambert Conformal Conic, although it might be Equidistant Conic.
Grab G.Projector (A free program made by NASA) and load a completely blank image into it. Turn on and configure the graticule, pick and configure a projection, and export the image.
NASA GISS: G.Projector — Global Map Projector



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