How do oxygen and carbon dioxide cross capillary walls?

The answer is diffusion.

The most important mechanism that enables oxygen and carbon dioxide (but as well other small molecules such as glucose, amino acids, wastes) across capillary walls is diffusion. Diffusion is a net movement of molecules through some barrier from an area of high concentration to the area of low concentration. When blood rich in oxygen reaches capillaries close to the cell, now there is more oxygen in the capillaries than in the cells and by diffusion, oxygen will pass capillary walls and enter the cell. Since blood in capillaries lacks in carbon dioxide, it will easily leave the cells and enter the blood. It should be taken into consideration that capillary walls may be fenestrated, continuous, and discontinuous which can affect movement through them.