The trick is to start perceiving mathematical objects as "real" as the ones in the so called "real world". In fact, they are way more real than the physical objects, but I'd rather not digress on that now and concentrate on one more thing that is rarely taught or even mentioned.
You can develop a dual vision in which you see abstract mathematical setups as common situations and vice versa easily translating from one language to another. Here is an example that I recently ran over with my students (it was pure improvisation: they asked me how to solve some problems from Analysis and I tried to explain a bit how one could visualize the situation).
Problem 1 (abstract mathematics): Given a countable set of sequences $a^{(n)}_1, a^{(n)}_2,\dots$ increasing to infinity, construct a single sequence $a_1,a_2,\dots$ still increasing to infinity such that it is eventually smaller than each sequence in the given countable family, i.e., for every $n$, there exists $K=K(n)$ such that $a_k\le a^{(n)}_k$ for all $k\ge K$.
The solution for the finite set of sequences would be trivial: just take $a_k$ to be the minimum of $a^{(n)}_k$ over all $n$. The minimum of finitely many growing to infinity sequences still grows to infinity and we are done.
However the countable case was not so easy for them. And then I restated the problem in the following way.
Problem 2 (traveling on a railroad) You have a railroad going from your current location $X$ to infinity with infinitely many stations on the way. Every hour a train departs from $X$ and each train is slower than the previous one. You want to use these trains to go to infinity but you should eventually find yourself behind any particular train. How do you do that?
The answer came in about 3 minutes: you take the first train to the first station, disembark there, wait for the second train and take it to the second station, then wait there for the third train and take it to the third station, and so on. After that it took some time to convert it back into the abstract mathematical language (there are some pesky translation details that I'm intentionally skipping), but once the idea was there, it all was more or less routine. The main point, however, is that for a mathematician there is absolutely no difference between Problem 1 and Problem 2. Solving one of them is pretty much the same as solving the other and, because of that equivalence, they have equal "tangibility" and other properties, though Problem 2 uses next to no mathematical notions (the only one that is really there is forward infinity in time and space, which most people would, probably, consider "physical" too).
At more basic level of elementary calculus, there is no difference between a girl holding a dog on a leash with the dog holding a cat on a leash and a question how short the leashes need to be to ensure that the cat doesn't travel more than 10 yards away from the girl on the one hand, and the formal standard $\varepsilon/2$ argument that the limit of the sum of two convergent sequences is the sum of the limits (or that the sum of two continuous functions is continuous, or...) on the other hand.
At more advanced level, one of the proofs of the Ergodic Theorem is just the big yellow bulldozer pushing a huge sand pile that fills all holes in the road on its way. I really keep in memory nothing but that about the proof when I come to the board to teach it: the rest is just the translation of that image into a chain of formulas.
It won't be a big exaggeration to say that (almost) all mathematical concepts (definitely all basic ones and the vast majority of even the most sophisticated ones) can be translated into plain English and the underlying ideas explained on the setups having nothing to do with "mathematical abstractions".
Why isn't that done more often? The reason is the same that we don't normally accompany our typing in English by pictures except in the children books: we rely upon the idea that it is enough to type 5 letters "g-r-e-e-n" to bring in the reader's mind the pictures of the forest, the lizards, the traffic lights, the stale bread, the ecological party, etc., etc. What makes it "tangible" for us is just a huge association table in our brain that contains the "green" entry in many places. By itself, however, it is not even a certain wavelength range: all you saw when reading this passage was 5 fancy wiggles made out of a few dozen pixels on the screen: pure intangible meaningless abstraction, as it is. The same happens with mathematics: as long as the long association chains are not present, you just feel like the book author juggles the same 23 symbols over and over again in various combinations for no apparent reason: hdudfhwuio jddljk hjscf kkfelr.
What to do to develop those association chains? My advice would be to relax a bit on studying the theory and pick some problem solving books. I don't know what your favorite area of mathematics is so I'll abstain from naming any titles for now, but trying to figure things out and to navigate that parallel Universe by yourself is a lot of fun even if the solutions of some particular problems are already known to humanity and you can pretty easily take a sidestep at any point to find yourself on a totally uncharted territory.
Finally, before trying to answer the question "What is a pure mathematics good for except producing more pure mathematics?" try to answer the question "What are human beings good for except producing more human beings?" It is actually the same question in the sense I tried to describe. Once you find a satisfactory answer to either version, just translate it into the other setup.
Just my two cents.
