How to display only a left and bottom box border in matplotlib?

Just set the spines (and/or ticks) to be invisible.

E.g.

import matplotlib.pyplot as plt

fig, ax = plt.subplots()

ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)

plt.show()

If you want to hide the ticks on the top and left as well, just do:

ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')