I want to verify if my intuition is correct:
Intuitively, I view "attaching an $n$-cell to $X$" as treating the $n$ dimensional hemisphere $S$ as rubber, you deform $S$ and attach $S$ to $X$ so that its boundary completely touches $X$, where the shape of $S$'s boundary (i.e. where it touches $X$) is determined by the attaching map $f$.
In particular, we're not just drawing the image of $S$'s boundary onto $X$, we are also attaching the rest of the hemisphere.
Also, why are we attaching hemispheres instead of spheres?
Is my intuition correct?
p.s. This is the definition of cell attachment I'm following:
An $n$-cell by definition is a topological space that is homeomorphic to the $n$-disk $D^n$, i.e the topological space that is bounded by the $(n-1)$-sphere $S^{n-1}$. Dimension $2$ resp. $3$ is where the intuition is coming from, that's why one imagines an $n$-disk to be the $n$-dimensional analogue of the $2$-disk which is up to homeomorphism just a hemisphere of the $2$-sphere $S^2$. Attaching a $1$-cell to $X$ is up to homeomorphism the same as attaching $D^1=[0,1]$ along its boundary $S^{0} = \{0\}\sqcup\{1\}$ to $X$, i.e. you glue something that looks topologically like the $[0,1]$ intervall along its two boundary points onto $X$. The attaching map $f\colon S^0\to X$ captures this attachment. You can try to visualize the same for dimension $n=2$, i.e. attaching $D^2$ along its boundary circle $S^1$ via $f\colon S^1\to X$.
In order to understand what a cell-attachment topologically actually is, you'll need to understand the concept of quotient topology and quotient spaces since gluing two topological spaces $X,Y$ along $A\subset X$ and $B\subset Y$ by a map $f\colon A\to X$ means that $$f\colon A\xrightarrow{\cong} \underbrace{f(A)}_{=:B}$$ is a homeomorphism and we identify $X$ and $Y$ along $A$ and $B$ respectively by defining an equivalence relation $\sim$ on the disjoint union $X\sqcup Y$ via $$x\sim y :\iff y = f(x)$$ The quotient $X\sqcup Y/{\sim}$ is naturally equipped with the quotient topology and thus you'll obtain a new topological space which we denote by $X\cup_f Y$ in order to emphasize that $X$ and $Y$ aren't identified under some arbitrary equivalence relation, but by the very explicit equivalence relation $\sim$ determined by $f$.
Intuitively and very roughly speaking: you take your space $X$ that contains some "subregion" $A\subset X$ (careful, this is really bad terminology in this context but it might help with intuition) and you glue $A$ onto some "subregion" $B\subset Y$ that looks topologically exactly like $A$, so you can just glue $A$ and $B$ together and you will have a new space, which we call $X\cup_f Y$, where $A$ and $B$ are glued to each other and the complements $X\setminus A$ and $Y\setminus B$ are unaffected by the gluing procedure.