Why not j = i + 1 ?

If you run the code like this: for (int j = i ; j < array.length; j++) it compares the element with itself in the first round of the loop. As you wrote: array[0] with array[0]. Then it compares these: array[1] with array[0] array[2] with array[0] array[3] with array[0] array[4] with array[0] outer loop increments i by 1: array[1] with array[1] array[2] with array[1] etc. We can see that the occasionally self-comparison does not cause an error. It is just a redundant operation. But the rest of loop still works fine.

Downloading a correct solution doesn't mean it is the ultimate correct solution, it is only just a solution. In this task the downloaded solution uses a selection sort algorithm which is one of the easiest to understand. You may have reached a different solution, maybe a bubble sort algorithm which a lot of new learners intuit on their own. I think, for small data sets such as this task, that either of these two algorithms would be the most efficient in terms of keeping the code brief, though a comparison sort would probably beat it out in terms of processing speed. When I first started on codegym, back in 2018, they didn't have the feature to download a solution. I believe, when they implemented it, that they just took passing code that students have previously submitted and used that as the code you can download when you want a correct solution. This code isn't optimized, as you have noticed, and is just 1 of potentially millions of different ways to solve a task.