I did not know much about micellar water, just that people use it took clean their faces and remove makeup. Knowing that behind every cosmetic product there is a science, I researched the ingredients of micellar water.
Let’s take a look at the ingredients in Garnier’s Micellar Cleansing Water:

Water, Hexylene Glycol, Glycerin, Disodium Cocoamphodiacetate, Disodium EDTA, Myrtrimonium Bromide, Poloxamer 184.
These words mean nothing to me and probably don’t mean much to you either, so lets break this down into scientific terms we might recognise.
Hexylene Glycol
Nomenclature: 2-Methylpentane-2,4-diol.

I have drawn what the displayed formula may look like. (Feel free to comment if you think I have drawn it wrong as I am still an amateur). It is within the functional group of alcohols, which can be identified by the ‘ol’ of the nomenclature and OH groups on the displayed formula.
Glycerin
Also called Glycerol. Glycerin has antimicrobial and antiviral properties as well as being a humectant. Humectants are types of moisturising agents that pull water into the outer layer of the skin from the deeper layers of skin or from the air, therefore, it is used for hydration.

Disodium Cocoamphodiacetate
It cleans the skin and hair by helping water to mix with oil and dirt so that substances can be rinsed away.
Disodium EDTA – Disodium Ethylenediaminetetraacetic acid
Binds to metal ions, deactivating them which acts as a preservative for cosmetic products and helps prevent metals from going into hair or skin.
Myrtrimonium Bromide
Acts as a preservative
Poloxamer 184
A synthetic polymer that is a cleansing agent which has the same function as Disodium Cocoamphodiacetate.
I want to focus on Hexylene Glycol and Poloxamer 184 because they are emulsifiers and surfactants. Nature has its own emulsifiers such as phospholipids which bridge the gap between lipids and water. I will be using phospholipids as an example of how Hexylene Glycol works as I am more familiar with this molecule. (Poloxamer 184 is a commonly used nonionic surfactant too).
Phospholipids are lipids which have hydrophilic heads (attracts water) and hydrophobic tails (repels water). The heads, in the case of phospholipids, have a negative phosphate group attached to the base glycerol molecule which makes them hydrophobic whilst the fatty acids tails have no charge. Here is a drawing of a phospholipid in full.

It is often abbreviates as this with just the head and tail.

Surfactants dissolve in water to mix lipids and water together and they do this because of the hydrophobic/hydrophilic structure. They can either form bilayer (as seen in the plasma membranes of cells) or micelles. Micelles look something like this:

Here the hydrophilic heads are attracted to the water so point out towards it and the hydrophobic tails form a core to get away from the water. The heads are water soluble and the tails are oil/lipid soluble. Due to this, the molecules will attract dirt and oil from your face and break them down into emulsion droplets.
To use micellar water you pour some onto a cotton wool pad. When this happens the surfactant will rearrange its structure. Cotton wool is made from cellulose which is hydrophilic, therefore, the heads will interact with it (like interacts with like). This forms an orientated monolayer.

The tails are now free to attract oil from the skin so that makeup is removed from your skin. Not all the dirt will be removed but most of it will. The oily base of makeup also plays a role in this.

I have not discussed the benefits and risks of using micellar water. I myself do not use it so if you want to use it I suggest you research the chemicals and ingredients that go into it beforehand. All cosmetic products will have different benefits and disadvantages which change depending on the type of skin you have and your allergies too.