Let's try to understand this intuitively . Let's say you are sitting in a car somewhere very close to the north pole, may be just 50 meters from the pole. That means you are at very high latitude, probably around 89.9 degrees north. The pole is marked by a flag that you can clearly see.
Now if I ask you to circle the pole in your car, first you will need to turn at right angles from the pole. Then , as you start circling the pole keeping the flag in sight, you can clearly feel that you will need to constantly turn the steering wheel towards the pole to be able to drive in a circle. If you try to drive straight, you will drive away from the pole.
Now I ask you to repeat the same experiment when you are sitting in your car at the equator, which is 0 degree north. If I ask you to circle the north pole, you can intuitively feel that all you need to do is to keep the steering wheel straight, and you will be able to circle the pole even when driving perfectly straight.
So what is different between these two situations, apart from the latitude which is just a number. If you think closely about these two situations, you will arrive at the solution.
When you are walking along the smaller red circle, you are NOT walking in a straight line. OTOH, when you walk along the equator, you are walking in a straight line.
One way to understand this is following. Assume the sphere in your picture is earth and the smaller red circle is a latitude line. You are walking along this latitude. At any point on this latitude line you can imagine a tangent plane. The normal to the tangent place is line from the center to the earth to the point where you are standing. In fact your upright posture IS the normal to the tangent plane, because you feet point directly to the center of the earth.
Now, when you walk/run/drive along this latitude line, you experience a centrifugal acceleration which will be along a horizontal direction, i.e. from the center of the latitude circle to the point where you are standing. So, as you can see, there is a non-zero angle between the centrifugal acceleration vector and the tangent plane's normal vector. It is this non-zero angle that determines that you are not walking in a straight line.
When you walk along the equator, (or any great circle on earth for that matter), this aforementioned angle vanishes. That is why walking on any great circle on a spherical surface is considered walking in a straight line on a curved 2-D surface embedded in an ambient 3-D surface.
When you drive along a high latitude, the tangent plane of the instantaneous location constantly tilts towards the north pole.