In this image, a linear regression fit is made to the data points $X\in \mathbb R^{N\times 2}$. As you can see, the fitted values $\hat y$ is inside a plane/hyperplane.
However, in linear regression, we can make the fit non linear by adding a polynomial basis. In my attached picture, if $X_2=X_1^2$, I would expect to have a curve when plotting the predictions in a two-dimensional space.
However, I don't understand how a plane in a 3-D space would make a curve in a 2-D space. The projected data has to be a hyperplane since in $\mathbb R^N$ we are projecting to a $\mathbb R^M$ space where $M≤N-1$
Once you define $X_2 = X_1^2$, the derivative w.r.t. $X_2$ is constant. Therefore, you see this "straight" plane in 3D. However, in 2D, where you have $X_1$ and $X_1^2$, your model is not linear w.r.t. $X_1$; therefore, you'll observe a curve (second-order polynomial). Let's take a simple example. Three random dots: In 2D you cannot connect the dots with one straight line, hence you'll have some non-linear "line". However, once you add another axis, you'll be able to fit a plane that goes through these points. The same principle holds for your example.