Can imaginary (illusionary) forces cause acceleration, and what exactly are they? We might be aware of illusion forces such as centrifugal force that doesn't really exist but we feel the force for sure, how is that exactly possible?
I don't yet know much in Physics but does any illusion force cause acceleration?
To anyone who answers, can you also please give me a reference to the topic?
 A: Fictitious forces seem to exist only from a Newtonian perspective: if we insist on thinking that the correct equation of motion is the Newton's second law:
$$m\frac{\text{d}^2x^i}{\text{d}t^2} = F^i$$
Then, we find that in a non-inertial reference frame we need additional terms. These additional terms are the "fictitious forces" which are not caused by any specific physical agent (a set of atoms identifiable as their source). But, if we assume that the correct equation of motion is given by:
$$m\left(\frac{\text{d}^2 x^i}{\text{d}t^2} + \sum_{j,k} \Gamma_{j k}^{i} \frac{\text{d}x^j}{\text{d}t} \frac{\text{d}x^k}{\text{d}t}\right) = F^i$$
where $\Gamma_{j k}^{i}$ are the Christoffel symbols associated to coordinates used for positions. Then no fictitious forces are necessary and everything is explained solely by real forces (caused by specific detectable concrete physical entities). In this formulation, the conjectured fictious forces appear as the additional terms, that mathematically come from the covariant derivative.
A: The fictitious forces appear in non-inertial frames of reference.  The forces can cause acceleration when viewed in such a frame.
Riding in a car that swerves left, my coffee and donuts slide across my lap to the right and spill all over the door.  From my reference frame of the car, I could say that a centrifugal force appeared pointing to the outside of the turn (the right).  This force acted on everything in the car.  The snacks didn't have enough friction to resist, so this force accelerated them to the right until they hit the door.
Meanwhile my office mates watching from outside have a different opinion.  From the nearly-inertial ground frame, they see the car turn left, but my coffee and doughnuts continue at the same speed.  The snacks do not accelerate (until the door intersects their path).  There is no force or acceleration that appears in this frame.
A: So called 'fictitious forces' are actually real. When you are in a car swering around a bend you definitely feel a force - the centrapetal force. Likewise gravity is not a 'fictitious' force, it's a real force that we feel every moment of our waking and sleeping lives.
The mistake of calling these forces 'fictious' follows from taking inertial frames to be the natural frame. This is the point of view of special relativity, where the special refers to the inertial frames which are special amongst all frames in that the laws of motion take the form explicitly written down in Newtin's Principia.
However, general covariance states all frames are natural. In arbitrary frames we  will find forces that are not apparent in others. In the natural terrestrial frame we find gravity. And in natural rotating frames we find centrapetal and centrifugal forces not apparent in inertial frames.
A: Dynamics is the extension of statics by inertial force F_inert=-dp/dt (minus!). A fundamental law of statics is the equilibrium of forces.
Sum over all external forces = 0
(Sum over all internal forces = 0)
Thus, a fundamental law of dynamics reads
Sum over all external plus inertial forces = 0,
see Newton's 2nd axiom.
Example 1, spring: If you press or stretch a spring of its equilibrium state to another position, it exerts a real counter-force of equal strength but opposite direction upon you, cf. Newton's 3rd axiom -- no fictitious forces involved.
Example 2, car changing direction: The car behaves as if a centripetal force acts upon it. Correspondingly, there is a counter-force acting in opposite direction, the centrifugal force. This is really felt by all being moving together with the car. -- For all outside the car, there is no change of motion and hence no counter-reaction. For this, centrifugal force is not a fictitious force.
Example 3, winds: Since the Earth's body and its atmosphere are not tightly bound to each other, the Earth's body moves relative to its atmosphere. For this, a cloud, which is at rest relative to the fixed stars (assumed to build an inertial system), moves relative to the Earth, although there is no (net) force acting upon it. From the point of view of a point fixed on the Earth's surface, the motion of the cloud relative to it is explained by a fictitious force. This fictitious force just compensates for the apparent (!) inertial force corresponding to the apparent (!) motion of the cloud.
Summa summarum, fictitious forces are forces which appear when changing from an inertial frame to a non-inertial one. Unfortunately, this is not always properly described in the literature.
Hope this helps -- if not, say it :-)
A: The centrifugal force exist - it is the equal and opposite counterpart of the centripetal force.  For example, imagine a ball attached to the end of a pencil with a string.  Hold the pencil and spin the ball in a circle.  The centripetal force from the string on the ball is what causes it to move in a circle.  The centrifugal force is the force of the string on the pencil which has the effect of trying to pull it out of your hand.
The paradox of centripetal v. centrifugal force is merely due to people having a poor intuition of force and inertia.  For example, when sitting in a car that is turning sharply, people will suppose that they feel a force "pushing" them in the opposite direction of the turn.  The reality is that they are being pushed- just not in the direction that they think.
Try this - have someone push on your hips from your left side (so they are pushing you to the right.  What happens?  Your hips move to the right, and your body bends.  Now which way is your torso leaning?  To the left.  This is exactly what is happening in the car.  The seat primarily pushes your legs and hips in the same direction as the turn since they are the most secured to the vehicle.  Your torso leans in the opposite direction.  This causes people to in
