Strang's proof of SVD and intuition behind matrices $U$ and $V$
Apologies that this turned out a little long.
I haven't seen the lectures but I've just taught through the book. So I understand what you're feeling. Strang is going for a more intuitive than axiomatic style in his exposition. So you're right; he assumes that the SVD exists and then derives what the data have to be.
But if you read the section backwards you can get a more deductive version. First, $A^TA$ is symmetric and positive semi-definite (previous two sections of the book). Therefore $A^TA$ is diagonalizable by an orthonormal matrix, and its nonzero eigenvalues are all positive. This is the key fact that allows the SVD to happen. Order them as $\sigma_1^2 \geq \sigma_2^2 \geq \dotsm \geq \sigma_r^2 > 0$. Notice $r = \operatorname{rank}(A^TA)$, which is equal to $\operatorname{rank}(A)$ (this is proven in Chapter 3 somewhere). Let $v_1, \dots, v_r$ be an orthonormal set of eigenvectors for these positive eigenvalues, and $v_{r+1}, \dots, v_{n}$ an orthonormal basis for the zero-eigenspace, i.e., the nullspace of $A^TA$.
Then he shows that if $v$ is a unit eigenvector of $A^TA$ with eigenvalue $\sigma^2$, then $u = \frac{1}{\sigma}Av$ is a unit eigenvector of $AA^T$ with eigenvalue $\sigma^2$. This is the key relation in the SVD. So if $V$ is the $n \times n$ matrix whose $i$th column is $v_i$, $V_r$ the first $r$ columns of $V$, $\Sigma_r$ the $r \times r$ diagonal matrix whose $i$th entry is $\sigma_i$, and $U_r$ the $m\times r$ matrix whose $i$th column is $u_i = \frac{1}{\sigma_i} A V_i$, we have $$ U_r = AV_r \Sigma_r^{-1} \implies U_r \Sigma_r = AV_r $$ Multiplying both sides by the transpose of $V_r$ and noting its columns are orthonormal, we have $$ U_r \Sigma_r V_r^T = A V_r V_r^T = A I_r = A $$
But wait, there's more! as Strang might say. The vectors $v_{r+1},\dots,v_n$ span the nullspace of $A^TA$. But the nullspace of $A^TA$ is the same as the nullspace of $A$. The vectors $u_1, \dots, u_r$ are $r$ (remember, this is the rank of $A$) orthonormal vectors in the column space of $A$, so they span the column space (a subspace of $\mathbb{R}^m$). We can complete the set $u_1, \dots, u_r$ with orthonormal vectors $u_{r+1},\dots,u_{m}$ to create a full orthonormal basis of $\mathbb{R}^m$.
We now have $r$ triples $(v_i,u_i,\sigma_i)$, where $Av_i = \sigma_i u_i$, and $n-r$ vectors $v_{r+1},\dots v_{n}$, where $A v_i = 0$. So if we let $\Sigma$ be the diagonal matrix $\Sigma_r$ augmented by $n-r$ columns of zeros and $m-r$ rows of zeros, and $U$ be the full $m\times m$ matrix whose $i$th column is $u_i$, it's still true that $AV = U\Sigma$. So again, $$ A = U \Sigma V^T $$ but now $U$ is an orthogonal $m\times m$ matrix, $V$ is an orthogonal $n\times n$ matrix, and $\Sigma$ is the sparse $m\times n$ matrix whose $(i,i)$-th entry is $\sigma_i$, with all other entries zero.
The final beautiful fact comes from taking orthogonal complements. We have an orthonormal basis $u_1, \dots, u_m$ of $\mathbb{R}^m$, the first $r$ of which span the column space of $A$. Therefore the remaining $m-r$ vectors $u_{r+1},\dots,u_m$ span the $C(A)^\perp = N(A^T)$. Likewise, $v_1,\dots,v_n$ is an orthonormal basis of $\mathbb{R}^n$, the last $n-r$ of which span the nullspace of $A$. Therefore the first $r$ of them span $N(A)^\perp = C(A^T)$. Thus the SVD produces not just the singular values and this nice factorization, but simultaneously a set of orthonormal bases for the four subspaces.
You're right that there's something wrong with assuming that we can do that. But there's nothing wrong with saying "If we could do that, what would it tell us?"
For instance, you might think, "I bet every square matrix is the sum of a symmetric and skew-symmetric matrix. Let's see what that tells us. If I write $$ M = S + A $$ where $S$ is symmetric, etc., then taking transposes I find that $$ M^T = S^T + A^T = S - A $$ and that tells me that $$ M + M^T = 2S $$ so $$ S = \frac{1}{2}(M + M^T) $$ and (similarly) $$ A = \frac{1}{2} ( M - M^T)." $$
OK, so having discovered that, you can ask yourself, "Suppose I define $S$ and $A$ by those last two formulas...will $S$ always be symmetric? Will $A$ always be skew-symmetric? Will their sum always be $M$?"
The answer to all three is "yes".
Now in writing up your proof that every matrix decomposes like this, you have two choices.
You can admit to the reader that you arrived at this idea by doing the "what if" exercise, or
You can pretend that the formulas were handed to you from heaven, or pure brilliance, or whatever.
I believe that Strang is taking the former approach: showing how, if you suspected that there might always be a singular value decomposition, you could arrive at what the matrices HAVE to be; then you work through a proof that those matrices actually DO have the properties that you want.
If you start from a false conjecture, like "5 is the sum of two adjacent even numbers", you can do the same thing. But when you find formulas for the even numbers, you'll discover...that they're not even. So assuming something false in order to conjecture a formula is no sin...but failing to check that the resulting formula is valid? That's bad.
I haven't answered the second part of your question ("Is there some intuition...") because one person's intuition can be another's bafflement. I don't see any immediate intuition in this case, but I don't have a great intuition for the meaning of the transformation represented by $A^t$ (except if we think about dual spaces, and I'm not certain that's useful here).