As another version of HJRW’s answer: free groups are (isomorphic to) fundamental groups of graphs and the rank is captured by the number of “independent” loops in the graph. For example $F_2$ is isomorphic to the fundamental group of the rose with two petals. Now use the correspondence between subgroups and covering spaces.

As another version of HJRW’s answer: free groups are (isomorphic to) fundamental groups of graphs. The rank is then captured by the number of “independent” loops in the graph. For example $F_2$ is isomorphic to the fundamental group of the rose with two petals. Now use the correspondence between subgroups and covering spaces.