What forces are at play when molecules wiggle (due to heat)?
Or in other words, What makes them move?
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Suppose your molecule is in a gas. This could be a gas made up just from the molecules you're thinking of, or your molecule could be dispersed in some carrier gas. Whichever the case, the temperature of the gas is related to the velocity of the gas molecules. So in a cool gas the molecules will be moving at some speed (that depends on their mass) and as you heat the gas the molecules will move faster. For any particular gas and temperature the velocities will be given by the Maxwell Boltzmann distribution.
The point of all this is that when you have gas molecules buzzing around in random directions they will collide with each other. For simple atomic gases like helium, neon, etc the collisions are elastic just like (idealised) pool balls. However when you collide molecules some of the collision energy can go into making the molecules vibrate and/or rotate. This is why if you could watch your molecule with a super powerful microacope you would see it "wiggle". All you're seeing is the kinetic energy of the colliding molecules being turned in intramolecular motion.
If you are looking at a liquid not a gas the same reasoning applies. The molecules in a liquid move much more slowly than in a gas, but then because they are so close they collide much more frequently. The collisions transfer energy to intramolecular motion in just the same way as in a gas.
If you have a relatively large particle of molecule in a liquid the collisions from the molecules of the liquid cause the phenomenon known as Brownian motion.
Response to comment:
If you have an isolated gas then it contains some finite amount of energy, and this energy is just the total kinetic energy of the atoms, plus any energy stored in intramolecular vibrations and/or rotations. Now energy conservation is a fundamental law in physics, so if we don't take energy out of our gas, or put energy in, the energy can't change. Within the gas some of the kinetic energy can be changed into molecular vibrational energy, or conversely vibrational energy can go back into kinetic energy, but the total amount of energy is constant. That means our gas molecules will go on buzzing around and vibrating forever.
You ask "how does heat exactly make molecules move?", but the point is that heat is molecular motion. Heat is the same as internal energy, and internal energy is kinetic and vibrational energy of the gas molecules. There isn't any mystical fluid called heat in a gas. There is just the energy of the gas molecules.
Finally, you ask "why doesn't a system - gas, liquid, solid - come to a rest?". The only way the gas molecules can come to a rest is if you cool the gas, i.e. you take energy out of it. Suppose you pump your gas through a cold pipe. The gas molecules hit the walls of the pipe and transfer some of their energy to it. The energy and therefore velocity of the molecules is reduced and the gas temperature falls. At the same time the cooling pipe heats up because it has absorbed energy from the gas. If you keep up the cooling for long enough the gas molecules will move so slowly that they stick together and form a solid. But it's important to note that the energy originally in the gas hasn't disappeared, it's gone into heating up whatever cooling system you're using.
Response to response to comment:
Vibration is motion too. If you've ever seen a vibrating guitar string you'll have seen that it's moving. If you stop the guitar string moving than it can't vibrate.
Take your example of a solid and for simplicity make it a simple atomic solid e.g. frozen argon. At absolute zero the Argon atoms don't move. The argon atoms are as close as they want to be i.e. if you try and push them together they want to spring apart. In effect they they behave as if there are tiny springs between them. The "springs" are actually the interatomic force; let's not go into the details here or this answer will expand to book length!
Anyhow imagine starting to heat the argon by putting it in contact with the hot gas we started with in the discussion above. The hot gas molecules collide with argon atom at the surface of our solid argon, and they transfer some of their energy to the argon atoms by making them vibrate. The gas molecule bounces off with reduced energy so the gas cools by making the argon atoms vibrate. The vibrating argon atoms at the surface push on the atoms the next layer down and make them vibrate, and so on until the whole block of argon heats up.
If you've ever held onto a metal spoon and dipped it into a hot liquid you'll know heat conducts up the spoon. Well heat conduction is just the atoms passing their vibrational energy from atom to atom along the spoon. And, as I said above, vibration involves motion.
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View the molecules as a collection of masses attached together with springs. The masses are the atoms and the springs are the interatomic forces from the bonds. An assemblage like that will have resonances and will wiggle in characteristics ways. The excitation of this assemblage will be either collisional, - other molecules hitting it or they can interact via van der waals forces over short ranges.
The various forces are all manifestations of how electrons form bonds.
I will add to your chosen answer that all physical observations until the surprising discovery of radioactivity and the table of elements in the 1800's, ( i.e. quantum mechanics) can be described perfectly by electromagnetic forces and interactions together with Newton's law's of motion.
In your particular question if you mean "wiggle" the brownian motion
observed in microscopes and even in the dust particles in sunshine, it is the collisional reaction to the random motions of the molecules in the gas or liquid.
Every interaction, i.e. collision between molecules, is electromagnetic; it is electromagnetic forces that mediate in gas, liquid and solid matter. John Rennie has given a comprehensive descriptive answer on how temperature is the kinetic and vibrational energy of molecules. He forgot to state that the forces mediating the collisions are electromagnetic. All neutral atoms and molecules have a fringe electric field ( and a bit magnetic) which creates the so called Wan der Waals forces , and those forces mediate in gasses and liquids and solids, though to understand solids quantum mechanics is needed. In gasses at collisional contact, the electron clouds of the colliding molecules repulse and give the billiard ball like scattering. In liquids there exist also attractive Wan der Waals forces, which form the fluidity. At a specific higher temperature the kinetic energy of the molecules in the liquid is so high that the attractive forces are too weak, and one goes into the gas phase.
In solid state the atoms and molecules are packed at the balance limit of their attractive forces from their fringe fields and quantum mechanical states, and the repulsive forces of the same charge electron cloud that surrounds each of them. In crystals the organization is almost perfect and the crystal symmetries appear. Crystals are good quantum mechanical demonstrations which show that matter is organized in quanta and not continuously. In a crystal/solid the only degrees of freedom that can have kinetic energy ( cf John's answer) are vibrational degrees of freedom, and the atoms vibrating collectively set the temperature.
Statistical mechanics also gives us black body radiation. A body radiates in electromagnetic waves its energy, and left in vacuum ( 0K outside temperature) it will radiate all the energy and cool down to 0K. Conversely, electromagnetic radiation can heat a mass and raise its temperature, by the radiation hitting the atoms and molecules and transferring their energy into the atoms kinetic and vibrational energy, again by electromagnetic forces. Here again enters the necessity of the quantum mechanical formulation, because experimentally black body radiation
does not follow classical physics, but quantum mechanical physics.