I'm reading Bott and Tu's book on differential forms in algebraic topology and they have the following observation made when computing the de Rham cohomology of $S^1$. They first conclude that $$H^1(S^1)= \operatorname{coker}(\delta)=\mathbb{R}$$ where $\delta$ is the difference map sending $(\omega,\tau)$ to $(\tau-\omega,\tau-\omega)$ from $H^0(U) \oplus H^0(V)$ to $H^0(U \cap V)$. It's worth noting that the cover $U \cup V$ is chosen so that the intersection will result in having a small interval containing the north pole and a small interval containing the south pole.
After this they find an explicit generator for $H^1(S^1)$ which is the reason for this post. They state that
We now find an explicit representative for the generator of $H^1(S^1)$. If $\alpha \in \Omega^0(U \cap V)$ is a closed $0$-form which is not the image under $\delta$ of a closed form in $\Omega^0(U) \oplus \Omega^0(V)$, then $d^*\alpha$ will represent a generator of $H^1(S^1)$. As $\alpha$ we may take the function which is $1$ on the upper piece of $U \cap V$ and $0$ on the lower piece Now $\alpha$ is the image of $( - \rho_V \alpha, \rho_U \alpha)$. Since $-d(\rho_V \alpha)$ and $d\rho_U \alpha$ agree on $U \cap V$, they represent a global form on $S^1$; this form is $d^*\alpha$. It is a bump $1$-form with support in $U \cap V$.
Then they have the following illustrations
Now I would appreciate if someone could help me understand what these illustrations represent as I do not see how to tie them together with the reasoning they gave.
The first image depicts a partition of unity subordinate to the open cover $\{U,V\}$. The functions $\rho_U$ and $\rho_V$ are chosen such that $\rho_U,\rho_V\leq 1$ and $\rho_U+\rho_V=1$, with support on $U$ and $V$ respectively.
The second image depicts the graph of a closed $0$-form (which is a function) $\alpha$ on $U\cap V$. Evidently $U\cap V$ is the disjoint union of two intervals around the poles, so we can take a function which is constant on the respective components as depicted.
The third image shows the graph of the function $-\rho_V\alpha$ and $\rho_U\alpha$. Since $\rho_V$ is supported on $V$, the function $-\rho_V\alpha$, which is a priori defined only on $U\cap V$, can be extended to all of $U$ by simply declaring it to be zero everywhere else. This is depicted by the graph. A similar image is given for $\rho_U\alpha$, extended as a function to $V$.
The final image shows that $-d(\rho_V\alpha)$ and $d(\rho_U\alpha)$ agree on $U\cap V$, which is required for them to glue into a globally defined $1$-form, as is explained in the paragraph you cited. The preceding images should help you convince yourself that these $1$-forms indeed do agree - but this is a matter of single variable calculus, which you should be quite familiar with.