There is a greater pressure in the vessels than in the fluid outside of them, and the fluid filters through the wall as fluid filters through a thin membrane outside of the body.

Osmosis takes place when two fluids of different osmotic pressure are separated by animal membrane.
Difference in osmotic pressure is due to differences in molecular concentration, the greater the number of molecules the greater is the pressure, and the greater rapidity of flow is from the fluid of less pressure to the fluid of greater pressure.

The molecular concentration of tissue and blood fluid is constantly being equalized by the process of osmosis.

In the injured tissue the conditions are more favorable for the fluid of the blood to pass from the vessels: by filtration, because owing to the dilatation of the arteries there is increased amount of blood and greater pressure within the vessels, and the filtering membrane is also thinner because the same amount of membrane (here the wall of the vessel) must cover the larger surface produced by the dilatation.

It is, moreover, very generally believed that there are minute openings in the walls of the capillaries, and these would become larger in the dilated vessel just as openings in a sheet of rubber become larger when this is stretched.