I’ll give it a try.
The simplest example that illustrates the basic problem of 'accidental life' is to understand what a protein is and how it is made. Search 'life of the cell' on YouTube for visual references to proteins. Without at least some grasp of proteins, a simple explanation is impossible. A protein in biology has little to do with the dietary term 'protein' so don’t think 'the stuff in meat'.
There are thousands of different types of proteins for doing different jobs in a cell. Anything that happens or gets done inside a cell is done either directly or indirectly by a protein. It is the most basic functional unit in a cell.
A protein is a molecular machine. I use the term 'machine' because of its interrelated combination of chemical, electrical and mechanical characteristics and the fact that it is very specific and functional.
A protein is made of amino acids. Amino acids are called the 'building blocks of life' for this reason. Making these 'building blocks' in the lab is as close to creating life as we have come, even though they can also form naturally. This is why one theory of life emerging is called 'protein world' since it seems logical that the 'simpler' protein came before the far more complex cell.
There are hundreds of different amino acids and each one comes in right and left handed versions (mirror images). Proteins only contain 20 of those and all are left handed. This creates a problem for naturally occurring proteins because if you mix in any of the other amino acids, or even a single right handed one of the 20, the protein is broken and will not function. And there is no mechanism in nature to prevent such contamination. But we are not yet to the real reason why biological life had to be designed.
A protein is a very specific chain of ordered amino acids between about 150 and 3500 long, depending on the protein. They do not function in this string form. In order to be functional, they must be 'folded' into a complex physical three dimensional shape, which is another barrier to 'natural' life forming. But we are still not at the crux of the problem.
Let’s say that in spite of the odds, the right order of only the correct amino acids does link up by chance. Let us further say that they accidentally fold into the correct functional configuration. If you are into math, the chances of that happening have been calculated at 1 in 10^77. For perspective, there are about 10^50 atoms in the entire planet of earth. But still, we are not at the bottom of the problem.
Remember that we are only talking about a protein so far. it takes hundreds to thousands of different proteins working in a coordinated fashion to make a single cell function. But for now let's ignore the mathematical improbability of that first protein and the hundreds of others needed.
You have probably noticed that I have not mentioned DNA yet. It is the nature of what DNA is that makes accidental life virtually impossible. Bill Gates compared DNA to a computer operating system, only DNA is far more complicated. It is the most complicated thing we know of and we have only begun to understand just how complex it is.
But it is NOT the complexity itself that explains why it had to be designed. It is the multiple hierarchical levels of symbolic representation in DNA that demands a design. DNA has a LANGUAGE with syntax, words, punctuation, definitions, etc.
Here is the breaking point. It is possible for a human mind to imagine something as complex as a protein forming as a result of naturally occurring chemical processes even if the odds are vanishingly small. Then multiply that by the thousands of protein types needed. Still you could say, well given enough time, multiple universes, etc. it could happen. It sounds desperate to me but You can’t say the odds are zero. I should add that even the 'evolution explains everything' crowd can’t defend this 'Protein World' scenario, so they usually default to something like 'RNA world' as a precursor to first living cell. RNA is basically half of a DNA strand.
But to accept that this happened by random chance you would have to believe the following:
By random linking up of nucleotides (the four molecules that are in DNA), a machine language containing the words, letters, syntax and punctuation necessary for defining all the needed proteins for 'life' came about. Notice that I said 'defining' the proteins, not the proteins themselves or even the amino acids needed to make a protein.
To over simplify, DNA is a ‘recipe', an ordered list of instructions and ingredients on how to build thousands of different proteins. DNA itself cannot do anything with these instructions. In order to be built, the DNA instructions have to be transferred to a Ribosome, which in turn is a very complex protein itself (hopefully you see the chicken and egg problem here).
The Ribosome reads the symbolic list of the recipe and begins gathering the required amino acids called for in the list. It assembles the amino acids into a string in the order specified in the DNA strand sent to it. (in the form of what’s called ‘messenger RNA')
After the amino acids are strung together, Some simpler proteins will spontaneously fold into their final three dimensional shape but most require yet other proteins to actively form them in the correct way. If they are not folded correctly they will not function and are often toxic.
Hopefully you followed that but to summarize, complex combinations of amino acids are possible given enough time and material. The odds are not what I would call possible but you can’t say that a protein by accident is impossible, in spite of its complexity.
What cannot be reasonably believed is that 'nature' took that first accidental protein and then invented a symbolic language (encoded in DNA) that was able to be read and executed by that protein in order to make more proteins.
A protein by accident - maybe.
A symbolic language describing all the needed proteins for life and simultaneously a molecular machine that understands that language and able to build according to the instructions - Nope.
It is the symbolic nature of DNA's language
that required 'design'.
That's as simple as I can make it.