TL;DR The kind of words you're talking about are defined not by their referent but by their function.
Meta-words
If we were to draw an extremely rough continuum of word concreteness, it might look like:
(1) Proper nouns, e.g. "the Statue of Liberty", where you could point at it directly
(2) Common nouns, e.g. "dog", where you could point at any dog as an example of the type
(3) Abstract nouns, e.g. "love", where you could describe some situation in which it's active
...
(n-1) Semantic operators, e.g. "the", that modify a set of referents
(n) Syntactic operators, e.g. "although", that connect units in the linguistic system itself
These last two sets contain meta items. There's no entity, concrete or abstract, to which "the" refers. Instead, it applies a rule to other elements. Similarly, no real-world entity corresponds to "although". Instead, it's the marker for a certain logical relationship between other elements.
I see in your profile that you're a JavaScript developer. In JS, the word window refers to a particular entity. So does window.document. But to what entity does the period . we see refer? None. Rather, it's an operator that applies the rule "the next item is an attribute of the former".
How do we know what these words mean?
I doubt there's a definitive answer to this question (no pun intended), but maybe someone else can speak to the psycholinguistic facts.
My two cents is that as children we learn words for the relationships we perceive between entities and between propositions much as we learn words for everything else we perceive.
We know intrinsically that there is a contradiction to the scenarios described by the propositions "Ahmed's car is in the driveway" and "Ahmed is not home".
The same way we gradually gather that "mom" refers to a person and "love" refers to a feeling for that person, we gather that "although" names the contradictory quality — though it takes longer cognitive development for us to perceive this relationship than something the eyes can see.
How we gather any of those facts is a good question for those who study language acquisition.
Once the child has gathered the function of such a word, they easily abstract that function and use the word freely. For example, they know that "but" lets them contradict something even when they don't know what their objection is or even have an objection in particular. "But — but — "
How do we define these words?
Luckily, this question can be answered since it's a matter of convention! Yes, those words certainly are troublesome to define, being part of the system that is itself used to define things.
In every language native speakers without grammatical or linguistic training find this difficult. The usual behaviour when asked to define "although" is to give synonyms, often tending towards an (apparently) more atomic synonym like "but". Once this atom is reached and its definition requested they give examples — contrastive examples with and without the word, if they have good instincts...
So what do dictionaries do? As noted in the TL;DR, they give not the referent but the function.
Non-meta words like nouns and verbs tend to be defined by synonyms. For example, the OED defines "ship" as "large sea-going vessel". You could substitute the one for the other in a given sentence and get comprehensible output.
But the difference in the entry for "but" is stark. Just about every one of the many definitions is a sentence or paragraph expaining the effect of using the word.
Here's the OED's account of the meaning most people probably think of first:
25. Introducing a statement of the nature of an exception, objection, limitation, or contrast to what has gone before; sometimes, in its weakest form, merely expressing disconnexion, or emphasizing the introduction of a distinct or independent fact, as the minor premiss of a syllogism: However, on the other hand, moreover, yet ...
So it's easy to perceive what these words mean but laborious to define them. Although language can be made to describe how it itself is used, that's not the purpose it's best adapted for.
