Explanation via Pictures

Simple Explanation by Analogy (via a story)
Freddie works at the Government Taxation Office (IRS/HMRC/ATO/CBDT etc). He deals with millions of people who come to see him everyday. And he has a very poor memory.
In a World Without Cookies:
One day a customer walks in to Freddie's customer care desk:
Customer 1: "Good morning Freddie, so did you change my address like I asked you to?"
Freddie: "I'm sorry. I don't remember who you are? Who are you?"
Customer 1: "Dude, I spoke to you last Monday regarding this issue! How could you forget!"
Unfortunately, the HTTP protocol is stateless. There is no way Freddie (the server) can identify different customers (clients) apart from each other. He doesn't remember. He has a very short memory. There is a solution though:
The World WITH Coookies:
The customer walks in to see Freddie (his name is Brian), but this time, the customer gives Freddie his taxation office ID card:
Brian May: "Good morning Freddie, My name is Brian May...so did you change my address like I asked you to?"
Freddie: "ah yes...hmmm......Brian May, Queen, Lead Guitarist, We Will Rock you......very interesting, I have your records here on my back end system.........let me bring up the records pertaining to your address........YES: I did in fact change your address. BTW since you gave me your ID that's all I need, you don't need to tell me your name is Brian May. Just give me your ID and I will be able to see that on my system".
Explanation of Analogy
You can think of a cookie as kinda like an ID card: if you identify yourself to the server, the server will remember who you are and will treat you accordingly:
e.g. it will remember what you've already ordered in your cart so far.
it will remember that you like reading your website in Tamil / Spanish / Cantonese / Swahili etc.
it can (basically) identify who you are.
In this particular case, it is the Government Taxation Office who issues out the ID cards.
Granted the analogy is a little strained and very simplified but hopefully, it will help you understand and remember the underlying concept.