The map you're asking about (and ones like it) is known as a portolan chart, and the lines on it are lines of bearing related to a compass rose.

The lines on it are windrose lines, that is, lines that are meant to be consistent with wind directions as related to points of a compass rose (of which there were usually multiple on a portolan chart). They are sometimes erroneously referred to as rhumblines -- technically rhumblines can only be drawn on maps using a modern projection; a rhumbline is one that passes through all meridians at the same angle. Rhumb lines have a constant bearing relative to true or magnetic north (or true or magnetic south) -- they wind up being spirals on a globe, but are straight on Mercator maps.

The idea is that it would give a navigator a consistent line of bearing to the landmark that they were trying to sail to, with allowances made for wind and current (and tacking into the wind, etc.)

The lines all over the map were meant to be copied from the compass rose nearest to the navigator, and extended out to wherever the navigator currently was by use of a parallel ruler to "walk" the bearing out, thus giving the navigator a rough compass course to follow. (I'm saying rough because while now we can steer a consistent course of, say, 98 degrees, our hypothetical navigator of the 16th century would put his helm east by southeast a half east and approximate it.)

The use of these charts was mostly while working inshore, to try to help navigators distinguish parts of coasts from one another as well as to help them approach a coast or a harbor from offshore, when combined with notes from a sailing manual or rutter. They were not well suited to open ocean navigation except as approximations, as the technology of the time and variations in magnetic declination in different parts of the world induced inaccuracies. (Magnetic declination is a fancy way of saying that the Earth's magnetic pole is different from its rotational pole -- that is, magnetic north is not always true north.)