Solve is often fine for polynomial problems and sometimes not aggressive enough for transcendental equations. And, if you look at the documentation closely, you might notice you were missing a set of { } around your list of equations. Reduce is sometimes better for transcendental problems, but gives solutions in a different form.

Simplify[ Reduce[{x1^2 + y1^2 == 6^2, x2^2 + y2^2 == 14^2, y2 == ((y1 - y2)/(x1 - x2))*x2, 
   0 == ArcTan[((y2-y1)/(x2 - x1) - y1/x1)/(1 + ((y2 - y1)/(x2 - x1))*(y1/x1))]}, {x2, y2}, Backsubstitution->True]]

yields, where && means And and || means Or

x1 != 0 && ((x2 == -((7 x1)/3) && ((Sqrt[36 - x1^2] + y1 == 0 && 7 Sqrt[36 - x1^2] == 3 y2) ||
 (Sqrt[36 - x1^2] == y1 && y2 == -(7/3) Sqrt[36 - x1^2]))) ||
 (7 x1 == 3 x2 && ((Sqrt[36 - x1^2] + y1 == 0 && y2 == -(7/3) Sqrt[36 - x1^2]) ||
 (Sqrt[36 - x1^2] == y1 && 7 Sqrt[36 - x1^2] == 3 y2))))

Removing that Backsubstitution->True will give you the results in a different form. Choose whichever better suits your needs.