Here's your (very simplified) explanation:
(You can skip the first paragraph if you watch this:
DNA/RNA came into existence billions of years ago, and we're not quite sure how because fossils never last that long. Our best research tells us that DNA is a chain-like molecule (an object so small that the individual atoms are visible when looking at the whole thing) that stores information in codes, much like morse code. Each link of the chain is made of a different chemical, and the whole thing's built so that it can be copied perfectly every time (this is how reproduction is possible). DNA can also be copied into RNA, which is also a chain-code but smaller and cheaper/cleaner to mass produce. The mass-produced RNA is cleanly and safely shipped to other machines in a cell, and tells them how to assemble more machines (aka proteins) by telling them which order to put each atom/molecule in. The resulting machines perform basically all the complex tasks in a cell, including determining which sections of the DNA to code each type of machine in the first place.
DNA can't learn or change (it actually can but lets not worry about that), it can only tell simple single-task molecule machines (aka proteins) to build a simple single-task machine that either 1: detects a problem/task, 2: solves the problem/task, or 3: find the section of DNA that tells you how to build and/or link together the first two machines. There are other parts of the cell that aren't protein-machines, but they're always built or absorbed into the cell by protein machines.
Again, we have no idea where the first DNA came from, only that it builds machines and copies itself. It can't learn or change intelligently because it's really small and just stores tons of information. Lots of people take that and insert god as an explanation, but that's premature because we barely have any idea where it came from. The important thing here is that DNA just stores information on how to make machines-that-do-things, it can't learn on it's own.
When any cell grows to twice in size, gets at least two copies of each machine, and splits into two smaller cells (reproduces), the DNA first makes a perfect copy of itself for the old cell and the new cell. However, for some reason, DNA always randomly makes mistakes in the copying process. They are completely random and uncontrolled. Usually these mistakes will happen on unimportant sections of the DNA (which is a thing) but sometimes the mistakes create random changes in the instructions for random machines. Usually the random changes cause the machines to break and do nothing, like taking a Honda Civic and giving it a giant cube of steel instead of it's engine block or one of it's tires. Sometimes it barely does anything, like missing one of the turn signals.
Over millions of years and trillions of replications, however, there are good changes every once in a while. They're random changes in instructions so they usually sabotage the machine it codes for, but in super-rare circumstances the change causes the machine to do something completely new, giving the cell a new capability.
As an example, if you twisted 20 honda civics in random directions, you'd probably waste 20 honda civics, but if you twisted trillions of honda civics, you'd eventually get one or two that, when turned on, would mash parts together in such a way that it can open a can if you shove a can into a specific spot. If you took the crude can-opening honda civic and made trillions of perfect copies, and randomly twisted each of them, you'd eventually get one that happens to have one of it's wheels turn and push dropped cans into the (by chance) undamaged opening spot, like a conveyor belt. If you dropped that into a swimming pool full of unopened cans at the right angle, it would open at least a few of them. It would be incredibly inefficient, but life isn't efficient at all. DNA-coded protein machines are also made of parts that are far more volatile and dynamic than the smooth and easily-predictable chunks of iron we use.
The imperfect copying is called mutation. When mutations randomly result in a cell having new abilities, or (more frequently) changing its inner structure or color randomly so it can escape or hide from predators better, that cell gets food, reproduces normally, and produces a new generation of mostly-identical offspring (babies). If it can get food or survive predators 1% more frequently than the ones without the mutation, than the mutant offspring will reproduce 1% faster than the originals without the mutation. After things even out after trillions of years, they'll be the only ones left because they'll eat more of the food than the originals, or 99% of the overall population will die from something random like a volcano or ice age, and the mutant ones will replenish the population faster after everything goes back to normal.
Sometimes the first mutant will just get eaten, or the first 100 offspring will get eaten by a single whale gulp, even if they're better. Nature doesn't care, the mechanisms aren't smart enough to do anything other than perform tasks, and the ones that shoot for that extra 1% growth/survival rate will often do so by killing or getting killed.
TL;DR dna tells cells how to do literally everything. It (mostly) can't learn on it's own, but it can copy itself and randomly make mistakes when doing so. A random mistake, every once in a while, might tell a cell to do a random thing that turns out to be good, like being longer and thinner so it can swim faster. Those cells end up eating more food and having more babies, and after trillions of years, they'll be the only ones left.
Evolution is an process of random machine creation that is ground against a pitiless world, because none of the tiny parts are complex enough to give a flying f**k. Evolution itself is endlessly more complicated than anything I described here, but this is all we know about the core process.
Go watch a video on youtube on it, this is my first shot so they'll be more efficient with your time. Comment if I got anything wrong, just in case.