In light of the edits made since I actually composed this reply, I will answer your question first and address your further requests regarding assisting laypeople in understanding philosophy later.
The problem is not that philosophy has no value in today's society, which would rather pertain to a different discussion altogether; the problem is that philosophy has so many branches that seem to be alien to one another, and that philosophy itself is not a science that can be neatly ordered into categories. Philosophy is also not universally underpinned by mathematics, and remains qualitative.
What does this mean? Allow me to explain step by step. The value of a subject has no influence on how necessary it seems to people. It is no challenge to explain sociology, psychology, number theory or English literature to laypeople, and yet these subjects are still viewed by many as dubious for their respective reasons—all of which apply to philosophy. Sociology and psychology might be treated with a degree of scepticism because they are not scientific, that is, quantitative in practice or underpinned by mathematics in theory. The subjects are also very accessible in a pseudoscientific way to pretty much anybody who is aware of a vague definition of the terms. Plenty of streetwise people will be very good at first-hand psychology for having met and interacted with so many people and having so much experience, though they would perhaps struggle to write an academic paper on behavioural trends they identify in people. Sociology and psychology also have a reputation for being 'soft sciences' because of their relative ease and accessibility compared to physics or chemistry—primarily because not a lot of mathematics or scientific method is involved in general practice. It is true that fairly advanced statistical mathematics can be involved in both subjects at an academic or 'research' level but when a subject's credibility is constrained to academia, this only serves to reinforce the scepticism of the layman, who is reminded of the practical applications and generalised, quantitative character of mathematics or physics every time he sits down to do some personal bookkeeping or talk about cars.
Despite the appreciation mathematics gets from laypeople, however, branches such as number theory or set theory which deal with very abstract concepts are also treated with suspicion and are difficult to explain or justify to laypeople because of the lack of obvious application in real life. It is true that one would rarely find themselves needing to derive the natural numbers from the cardinality of sets proceeding from the empty set, but such abstract investigation provides the foundation for everything that sits above in calculus, physics, chemistry, statistics and so on. This abstract character of pure mathematics precludes it from being accessible to the layperson.
English literature is likewise taken with a pinch of salt by the layperson because the acts of reading and interpreting The Odyssey or Don Quixote do not prove really useful in jobs that laypeople admire. It is true that while degrees like English literature are these days advertised as providing 'transferable' skills to the plucky graduate, no law firm is going to pass up a law graduate for an English literature graduate because no relevant knowledge will have been acquired during the latter's degree. While laypeople may concede that interpreting and understanding metaphors in classical literature, discerning the subtleties of social commentary in modern literature and deciphering the true meanings of obscure poems does require a talent they may not possess, they will not see much justification for a subject that exists for its own sake and does not really provide knowledge about practical things.
Each of these reasons why laypeople are bemused by certain subjects apply to philosophy: philosophy is a 'soft' subject to a layperson that requires little in the way of mathematical talent; philosophy deals with some thoroughly abstract concepts that cannot be easily explained ot discussed with laypeople (try talking about Dasein in a bar); and philosophy seems quite closed off and self-serving as an academic discipline, not offering much in the way of transferable skills or knowledge (do any science degree and it will be presumed that you have a better grasp of mathematics and logic than a student of the humanities).
These reasons share a common derivative: an emphasis on what is necessary to get a job with a decent salary. The focus on finding a job through academic study is simply a sign of the times. Gone are the days when university was a privilege reserved for those who displayed true talent; basically everybody now has a right to attend a university and as a result, the job market can pick from thousands of graduates who have years of experience working with computers and reading books. Degrees are fast becoming a minimum requirement for low level jobs where they would once have granted you access to top level jobs in research, teaching or in companies. The capitalistic society in which we live has mutated into a culture that does not value academics outside of academia, and academia has a limited number of positions available. Thus everybody is encouraged to take a practical degree like mathematics, physics, chemistry, statistics, business studies, law and so on, not simply because the best salaries lie outside of academia, but because the best chances of being employed at all lie outside of academia.
It is true that philosophy can be all but impossible to translate into simple English, especially when much of philosophy is barely translated into academic English without mistakes and vagueness. Philosophy itself even concerns itself with its abstruse nature and the difficulty or even potential impossibility of translating concepts between different languages without a good degree of abstraction and personal translation from the student.
tl;dr a simple analogy. Imagine computer programming languages. You have various kinds of programming languages and the high level languages as well as scripting languages tend to create a route to a high salary or good business opportunities. Certified competence in C++ or Java will net you a well-paid code monkey job quite easily; same goes for javascript in web development. Being comfortable with Go, Erlang, Clojure, Node.js, etc. would give you a great footing from which to create a website or application that could net you some money as a business venture if you have a good idea as well. No one really thinks about the low level side. You do not really see job advertisements for assembly language experts, or people who specialise in fundamental algorithms (even in spite of the fact that a good mathematical understanding is required to grasp fundamental algorithms, optimisation and computational complexity). Yet without people who could originally program in machine code, or those who developed the first single-function computers, and without computer scientists developing the first high level languages and platforms from the low level computation, none of it would exist. Same goes for philosophy. Philosophers do not get any credit for laying the foundations for the sciences and asking the questions that came to inspire some of the greatest pieces of literature. Philosophers are these days treated with a degree of disrespect even within their own sphere, mocked for how wrong they turned out to be or how silly their thinking now seems, yet this only indicates that the critical thinking that once was is now long gone. Science has won the battle and supplanted philosophy as our conceptual framework. And why not? Philosophy was originally around to logically and systematically explain the phenomena we now understand mathematically, physically and programmatically.
The great pity is that philosophy left room for art, whereas the mechanical framework that surrounds today's society leaves room only for producing things that sell. We as a society truly do know the price of everything and the value of nothing, and this is especially true of laypeople when they judge academic subjects. As we can see from the above examples, academic subjects are judged by the money that can be made and the probability of money-making, not by the value of the subjects. Philosophy is not the only victim though. Art is now based on what sells or shocks (and therefore sells) rather than encrypting and aesthetically presenting an idea or a feeling. Art has deferred to design and trendiness, while philosophy has deferred to science in the West (following the Analytic tradition) and where the continental tradition prevails, it is sorely misunderstood and barely valued by the current generation, who mostly use philosophical works for the sake of blog posts or to enshroud their personality in a haze of obscurantism.
How does one help non-philosophers access and understand philosophy? Do what current lecturers should be doing: scrap the classical thinkers in the beginning, and skip the Enlightenment entirely. Skip utterly unphilosophical questions about deities and abortion. Begin with the few things that are not in the domain of science. And remember: the Socratic method is the best way to teach everything, especially philosophy.
