This is my second answer, at the request of D.R.; for the beginning see the first one and the comments below : this one will explain how we can get the notion of compactness from the idea that a compact space is a space where you can't run of to infinity; "there's no infinity", or in other words "the space is 'finite' ". This answer is quite long too, so I wrote a (shorter and perhaps less informative than the first one) tldr at the end.
A first thing to understand is what it means to run of to infinity : notice that it can't be about distance or metric, because for instance $\mathbb R$ and $(0,1)$ are homeomorphic; and in fact morally it's quite clear that the sequence $1/n$ in $(0,1)$ "goes to (some) infinity".
Another example to see that distance and metric are not a reasonable way to formalize this "going off to infinity" (one I quite like, in fact analyzing this notion gives an elementary proof of a well-known theorem) is $\mathbb C^* = \mathbb R^2\setminus \{0\}$. In that space, you can go to infinity in two different directions : have your modulus increase to $\infty$, or decrease to $0$.
Note that in the "compact friends" of these spaces these issues are resolved : in $[0,1]$, $1/n$ gets closer and closer to $0$, not "far from everything"; in $D^2$ (the disk), if you get close to the boundary well you get close to the boundary, not far from things (same for the center); that's a good sign that our intuitive understanding has some connection to the notion of compactness.
In fact, in my examples I used sequences to exemplify running off at infinity, and that's why the BW property is relevant : if your sequence $(x_n)$ has the convergent subsequence $(x_{\varphi(n)})$, converging to $x$, then it can't be running off to infinity, because at least some of it is running towards $x$.
Let's look at an example where sequences don't do the trick (similarly to my other answer), which will explain the need for something different from BW : the space $\omega_1$, with again the order topology (or if you know what it is and like that better, you can use "the long line").
In this space, you can also run off to infinity, by picking larger and larger ordinals and never stopping. Of course if you do that following a sequence, then basic ordinal properties tell you that you're actually running towards something, and not infinity, but we do feel that there's room to run off to.
So if we want compactness to be something that a space has if "there's no way to escape to infinity", we must use something that's not defined by sequences. At this point I could branch out to my other answer and just say nets are the answer, but I'll try to go a different route and get to open covers more directly.
Picture $\mathbb R^2$ in your head. Intuitively, the only way to run off to infinity in this space is to have your norm increase a lot. This means that, as 'time' goes by, you'll be getting out of more and more balls of the form $B(0,r)$, $r>0$. One can even take that as a heuristic definition of running off to $\infty$ in $\mathbb R^2$ : for each $r$, you'll be out of $B(0,r)$ at some point (in 'time', but we don't want to specify what we mean precisely by time, that has a strong chance of being restrictive - except if we use arbitrary partial orders, but in that case we end up back to nets). But of course, at each fixed moment in time, you're in some $B(0,r)$ : the fact that you're able to run off is somehow related to the increasing 'sequence' $B(0,r)$.
What about, say, $(0,1)$ ? Well with a similar analysis, you see that you can only run off to $0$ and $1$ (anything inbetween is, well, in $(0,1)$ so you're running towards something, not away from everything); and so it has to do with getting out of the $(a,b)$'s, with $a,b\in (0,1)$.
But this example we knew how to explain with sequences; what about $\omega_1$ [you can again use the long line] ? In this one, sequences can't explain our feeling, but the idea is simply that if you give me $\alpha <\omega_1$, I can run off above it in a split second, but of course at any given time I am stuck below some $\alpha < \omega_1$. So I am always in some $[0,\alpha)$, but I can always go beyond these. So the problem is with the 'sequence' of $[0,\alpha)$'s. Starting to see a theme?
We see that we are able to run off to infinity when there are some subsets $X_i$ of our space that, in a sense, are everything at every given moment, but you can always go beyond them. What kind of subsets should we take ? Our previous examples suggest that they should be open, but isn't that just a coincidence ?
I could have take the closed balls in the first example, the closed intervals $[a,b]$ in the second and $[0,\alpha)$ is closed in $\omega_1$ anyway. But would these have been reasonable choices ?
Running away from everyone means that if $x$ is someone, at some point you'll be far from $x$. But neighbourhoods of $x$ (or open sets containing $x$) symbolize what is close to $x$. To be far from $x$, certainly you should be away from these.
So if your $X_i$ are your subsets that model the "running away", for each $x$ there should be some $i$ such that not being in $X_i$ ensures that you're far from $x$. The best way to do that is to make sure that the interiors of the $X_i$ cover the space.
To simplify things, we might as well say that the $X_i$ are open; and for that reason let's call them $U_i$. So now we have modeled a situation of "trying to run away" by an open cover $(U_i)$ of our space ! The question is now : in which of these situations can you actually run away ? What property of the space can we ask for to ensure that in every situation you can't run away ?
For instance look at the following in $\mathbb R^2$ : I take $U_0$ to be the right open half-plane, $U_1$ to be the left open half-plane, and $U_2$ to be an open vertical (infinite) band centered around $0$. By our previous analysis, these model a "situation of trying to run away", because they cover the space. But if you try to run away from them, since there are only $3$ of them, you will return to at least one of them infinitely often, so you haven't run away from any of the points that were in that one.
Ah ! This works because there are $3$ of them, but if there were only $4,5$, or actually any finite number of them, the same reasoning would apply : I wouldn't really be running away. So for a "situation of trying to run away" to allow you to run away you need to make sure that there's an infinite number of opens in your cover.
But there needs to be really infinitely many of them : take the same situation as above with my $3$ open sets. I can add as many as I want (an open ball here, an open square there, oh and perhaps some other open half-planes etc.), the reasoning I made will still apply to the original $3$.
So if I give you an open cover which models a "trying to run away situation", when can you ensure that I can't make this reasoning ? At this point, the notion of finite subcover arises, and you quickly notice that if from any open cover I can extract a finite subcover, then I can never really run away.
But wait, maybe that's too strong ! I made one reasoning that happened to use finiteness to get the result I wanted, maybe there are other ways to ensure you can't run away.
Well let's take a counterexample : an open cover $(U_i)_{i\in I}$ that has no finite subcover. So for any finite subset $J\subset I$, $(U_i)_{i\in J}$ doesn't cover the space so I can find $x_J\notin \bigcup_{i\in J}U_i$. Now what is my "process" $(x_J)_J$ ? Well it's something that is running away from my situation, as $J$ increases. Indeed let's take any point $x$ of my space : $x\in U_{i_0}$ for some $i_0$. Then for any $J$ that contains $i_0$, $x_J\notin U_{i_0}$ : $x_J$ is far from $x$.
That means that given any point, my running away takes me far from it if I wait "long enough"; in other words I successfully ran away from everything.
Therefore we have it : open covers correspond to "trying to run away", and those in which you can are precisely those that have no finite subcover. In particular to ensure that you can never run away, you have to impose this (strong-looking) condition : any open cover has a finite subcover. That is the definition of compactness.
I suggest you play around with various open covers and try to see what my story corresponds to with these, you should get a better feel of why I'm saying what I'm saying (hopefully).
In particular note that this interpretation does give two very different ways to run off to infinity in $\mathbb R^2\setminus \{0\}$, as our heuristic understanding suggested : ine one of them you take the open cover $(\{x \mid ||x|| >r\})_{r<0}$ and in the other $(\{x \neq 0 \mid ||x||< r\})_{r>0}$; and if you think about it, you'll see that morally (I don't have a precise statement in mind although I'm sure I could cook up some if asked to) any "essentially infinite" cover (that is, a cover of $\mathbb R^2\setminus \{0\}$ of which no finite subcover can be extracted) is basically one of these two, or a combination thereof.
If you formalize one (more advanced, that's why I said I didn't have a precise statement in mind) interpretation of this, you get an elementary (meaning : no algebraic topology involved) proof that $\mathbb R^2\setminus \{0\}$ and $\mathbb R^2$ are not homeomorphic.
$\mathbf{tldr}$ : Compactness of a space means you can't run off to infinity. When you think about it hard enough, you see that running off to infinity can be modeled by an open cover with no finite subcover, as can be seen in the examples of $\mathbb R^2\setminus \{0\}, (0,1)$ or $\omega_1$; the definition of compactness follows at once.