The words in the title may not be accurate, so I put them in quotes.
Recently I begin reading Hatcher's book on Algebraic topology. I have not studied any "topology" other than basic point set topology. I have much more experience with analysis-related topics (analysis of Banach and Hilbert spaces, for example).
I feel a lot of discomfort when reading sections of geometrical descriptions about how one space is homeomorphic to another. For example, the following paragraph taken from section 1.3 (Van Kampen's Theorem) is causing me some troubles. $K$ is a torus knot, embedded in $\mathbb R^3$. Please ignore $X$; it is not relevant to the discussion below.
It is very easy to understand why $\partial D^4 = \partial D^2\times D^2\cup D^2\times \partial D^2$. This is just an exercise of point-set topology. The part after "geometrically" is, however, difficult for me. Although it is quite natural to regard $\partial D^2\times D^2$ as a solid torus, I cannot see any obvious reason why it can be identified with the solid torus bounded by the torus knot $K$. It is also very difficult to see why the complement of this solid torus is homeomorphic to $D^2\times \partial D^2$. (It might be not as difficult as I think, but from time to time, I get stuck on some "obvious" things.) Whenever I come across such descriptions, I struggle a lot, thinking about how to prove those statements.
Also, the sets $D$, $S^3$ and so on in the book are not specifically defined (for example, as $D^2=\{(x,y):x^2+y^2\le 1\}$). Although defining $D^2$ specifically is clearly unnecessary, sometimes, the vague concept of $D^2$ causes me some difficulties. I find it really hard to tell whether the author is making a rigorous statement or a non-rigorous description.
In the end, in this book, I find it much easier to read proofs than to read remarks and other descriptive sections, due to the problems outlined above. Interestingly, when I read analysis books, the opposite is true: remarks are much easier to read than proofs. I am wondering why this is the case.
How can I overcome those problems/difficulties when reading this book?
This is not seeking personal advice - I believe that this might be the feeling of many other people as well. Please ask me to clarify if anything is unclear.
Well, your example is sort of hard to read without a blackboard because a lot of different images are going on. First, it's also worth noting that the identification $D^4=D^2\times D^2$ is only true up to homeomorphism when working with the $2$-norm. Only for the $\infty$-norm is it an on-the-nose identity of spaces.
However, Hatcher wants to map to $\mathbb{R}^3$, and the canonical way of going there is via the stereographic projection from the north pole $\phi_N((x,y,z,w))=\frac{1}{1-w}( x,y,z)$ (when realising $S^3$ under the $2$-norm), which has inverse given by $(x,y,z)\mapsto \frac{2}{x^2+y^2+z^2+1}(x,y,z,0)+\frac{x^2+y^2+z^2-1}{x^2+y^2+z^2+1}(0,0,0,1)$. Note that $\phi_N^{-1}$ extends continuously to the $1$-point compactification by mapping $\infty$ to $(0,0,0,1)$.
Okay, so what's $S^1\times S^1$ in $S^3$ under the $2$-norm? Well, it's any set of the form $\{(x,y)\in(\mathbb{R}^2)^2| \|x\|_2=r,\|y\|_2=s\}$ with $r^2+s^2=1$. For simplicity, set $r=s=\frac{1}{\sqrt{2}}$. For such an element $(x_1,x_2,y_1,y_2),$ we see that, for fixed $y_2$, the image of $\phi(x,y)$ is two circles for $|y_2|\neq \frac{1}{\sqrt{2}}$ and a unique circle for $|y_2|=\frac{1}{\sqrt{2}}$ (since this forces $y_1=0$). Now, at the same time, we see that, as $y_2$ ranges over $[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}],$ the third component of the image ranges over an interval (by continuity). Hence, $\phi(S^1\times S^1)$ is, indeed, the usual torus shape. Hence, $\mathbb{R}^3\setminus \phi_N(S^1\times S^1)$ has two components. Let $U$ denote the "inside", i.e. the bounded component, the closure of which is homeomorphic to the solid torus $\partial D^2\times D^2$. Hence, so is $\phi^{-1}(\overline{U})$. However, consider the stereographic projection $\phi_S$ from the south pole instead, and you'll see that it completely flips the picture, i.e. $$ \phi_S(\phi_N^{-1}((\mathbb{R}^3\cup \{\infty\})\setminus U)=\overline{U} $$
Hence, the outside of the torus (with $\infty$), is homeomorphic to the inside of the torus, which exactly gives you Hatcher's claim.
As for generally understanding intuitive explanations, I really recommend trying to draw situations. Naturally, drawing in four dimensions is pretty hard, but you can actually make a drawing in $3D$, where you end up cutting the top and bottom of $S^2$ (which gives you a copy of $\{0,1\}\times D^2$), which leaves you with a middle part (homeomorphic to $[0,1]\times S^1$).
As a finishing anecdote, a wise, older student once told me: "There are two kinds of people who do algebraic topology: The ones that love all the diagrams, and try to ignore the geometric pictures, and the ones that love weird geometry and decide to live with all the diagrams." I think that sort of explains the weird interplay between the abstraction and the handwaving.