Why do same/opposite electric charges repel/attract each other, respectively?
Well it has nothing to do with the Higgs, but it is due to some deep facts in special relativity and quantum mechanics that are known about. Unfortunately I don't know how to make the explanation really simple apart from relating some more basic facts. Maybe this will help you, maybe not, but this is currently the most fundamental explanation known. It's hard to make this really compelling (i.e., make it seem as inevitable as it is) without the math:
Particles and forces are now understood to be the result of fields. Quantum fields to be exact. A field is a mathematical object that takes a value at every point in space and at every moment of time. Quantum fields are fields that carry energy and momentum and obey the rules of quantum mechanics. One consequence of quantum mechanics is that a quantum field carries energy in discrete "lumps". We call these lumps particles. Incidentally this explains why all particles of the same type (e.g. all electrons) are identical: they are all lumps in the same field (e.g. the electron field).
The fields take values in different kinds of mathematical spaces that are classified by special relativity. The simplest is a scalar field. A scalar field is a simple number at every point in space and time. Another possibility is a vector field: these assign to every point in space and time a vector (an arrow with a magnitude and direction). There are more exotic possibilities too. The jargon term to classify them all is spin, which comes in units of one half. So you can have fields of spin $0, \frac{1}{2}, 1, \frac{3}{2}, 2, \cdots$. Spin $0$ are the scalars and spin $1$ are the vectors.
It turns out (this is another consequence of relativity) that particles with half integer spin ($1/2, 3/2, \cdots$) obey the Pauli exclusion principle. This means that no two identical particles with spin $1/2$ can occupy the same place. This means that these particles often behave like you expect classical particles to behave. We call these matter particles, and all the basic building blocks of the world (electrons, quarks etc.) are spin $1/2$.
On the other hand, integer spin particles obey Bose-Einstein statistics (again a consequence of relativity). This means that these particles "like to be together," and many of them can get together and build up large wavelike motions more analogous to classical fields than particles. These are the force fields; the corresponding particles are the force carriers. Examples: spin $0$ Higgs, spin $1$ photons, weak force particles $W^\pm, Z$, and the strong force carriers the gluons, and spin $2$ the graviton, carrier of gravity. (This fact and the previous one are called the spin-statistics theorem.)
Now the interaction between two particles with "charges" $q_{1,2}$ goes like $\mp q_1 q_2$ for all the forces (this is a consequence of quantum mechanics), but the sign is tricky to explain. Because of special relativity, the interaction between a particle and a force carrier has to take a specific form depending on the spin of the force carrier (this has to do with the way space and time are unified into a single thing called spacetime). For every unit of spin the force carrier has you have to bring in a minus sign (this minus sign comes from a thing called the "metric", which in relativity tells you how to compute distances in spacetime; in particular it tells you how space and time are different and how they are similar). So for spin $0$ you get a $-$: like charges attract. For spin $1$ you get a $+$: like charges repel! And for spin $2$ you get a $-$ again: like charges attract. Now for gravity the "charge" is usually called mass, and all masses are positive. So you see gravity is universally attractive!
So ultimately this sign comes from the fact that photons carry one unit of spin and the fact that the interactions between photons and matter particles have to obey the rules of special relativity. Notice the remarkable interplay of relativity and quantum mechanics at work. When put together these two principles are much more constraining than either of them individually! Indeed it's quite remarkable that they get along together at all. A poetic way to say it is the world is a delicate dance between these two partners.
Now why do atoms and molecules generally attract? This is actually a more complicated question! ;) (Because many particles are involved.) The force between atoms is the residual electrical force left over after the electrons and protons have nearly cancelled each other out. Here's how to think of it: the electrons in one atom are attracted to the nuclei of both atoms and at the same time repelled by the other electrons. So if the other electrons get pushed away a little bit there will be a slight imbalance of charge in the atom and after all the details are worked out this results in a net attractive force, called a dispersion force. There are various different kinds of dispersion forces (London, van der Waals, etc.) depending on the details of the configuration of the atoms/molecules involved. But they are all basically due to residual electrostatic interactions.
Further reading: I recommend Matt Strassler's pedagogical articles about particle physics and field theory. He does a great job at explaining things in an honest way with no or very little mathematics. The argument I went through above is covered in some capacity in just about every textbook on quantum field theory, but a particularly clear exposition along these lines (with the math included) is in Zee's Quantum Field Theory in a Nutshell. This is where I would recommend starting if you want to honestly learn this stuff, maths and all, but this is an advanced physics textbook (despite being written in a wonderful, very accessible style) so you need probably at least two years of an undergraduate physics major and a concerted effort to make headway in it.
An experimentalists answer:
Why do same/opposite electric charges repel/attract each other, respectively?
Because careful physicists have made an innumerable number of observations and have found that this is what nature does. There is a long history of observations before any theory could be solidified. They observed the behavior of attraction with some types , repulsion with others and defined the + and - to separate the two sets.
Classical electromagnetic theory modeled the behavior of charges very well , with Maxwell's equations. They show how, when positive and negative charges exist in nature they can be modelled with accurate mathematical solutions of the equations and one could think that your "why" would be answered by " because they fulfill Maxwell's equations".
Then quantum mechanics came as a revolutionary mathematical theory to describe phenomena measured in the microcosm, including charged elementary particles, and a theory was developed as explained in Michael Brown's answer above, which again models extremely well the behavior of charged particles, and your "why" can be answered by "because they fulfill quantum electrodynamical equations".
You must perceive then that the "why opposite charges attract" with the answer "because that is what we have observed" becomes "because we have modeled mathematically the observations successfully". Then the question becomes why this mathematical model, and the answer is "because it describes the observations", circular.
I am pointing out that "why" questions can not be answered with physics. Physics can be successfully modeled mathematically with postulates and using the model one can show how the behavior of positive and negative charges under all sorts of experimental conditions can be predicted accurately, but not "why" they exist. The why gets the answer "because that is what we have observed nature to do".