There are (at least) two ways to think about this.
One is as you mentioned: imagine the points being lifted into the shape of a quadratic function, and then being cut by a plane, producing an ellipse. This is kind of like this picture (stolen from this paper):
Another way to think about it is: the decision boundary for an SVM will always be of the form $\{ y \mid \sum_i \alpha_i k(x_i, y) = 0 \}$$\{ y \mid \sum_i \alpha_i k(x_i, y) = b \}$. For the kernel $k(x, y) = (x^T y + c)^2$, we have: \begin{align} \sum_i \alpha_i (x_i^T y + c)^2 &= \sum_i \left[ \alpha_i (x_i^T y)^2 + 2 \alpha_i x_i^T y + \alpha_i c^2 \right] \\&= \sum_i \alpha_i y^T x_i x_i^T y + \left( \sum_i 2 \alpha_i x_i \right)^T y + c^2 \sum_i \alpha_i \\&= y^T \left( \sum_i \alpha_i x_i x_i^T \right) y + \left( \sum_i 2 \alpha_i x_i \right)^T y + c^2 \sum_i \alpha_i \\&= y^T Q y + r^T y + s ,\end{align} which is itself a quadratic function. So the decision boundary is always going to be the level set of some quadratic function on the input space.