While the accepted answer is quite good, it still doesn't touch at the real reason why $O(n) = O(2n)$.
Big-O Notation describes scalability
At its core, Big-O Notation is not a description of how long an algorithm takes to run. Nor is it a description of how many steps, lines of code, or comparisons an algorithm makes. It is most useful when used to describe how an algorithm scales with the number of inputs.
Take a binary search, for example. Given a sorted list, how do you find an arbitrary value inside it? Well, you could start at the middle. Since the list is sorted, the middle value will tell you which half of the list your target value is in. So the list you have to search is now split in half. This can be applied recursively, then going to the middle of the new list, and so on until the list size is 1 and you've found your value (or it doesn't exist in the list). Doubling the size of the list only adds one extra step to the algorithm, which is a logarithmic relationship. Thus this algorithm is $O(\log n)$. The logarithm is base 2, but that doesn't matter - the core of the relationship is that multiplying the list by a constant value only adds a constant value to the time.
Contrast a standard search through an unsorted list - the only way to search for a value in this case is to check each one. Worst-case scenario (which is what Big-O specifically implies) is that your value is at the very end, which means for a list of size $n$, you have to check $n$ values. Doubling the size of the list doubles the number of times you must check, which is a linear relationship. $O(n)$. But even if you had to perform two operations on each value, some processing, for example, the linear relationship still holds. $O(2n)$ simply isn't useful as a descriptor, since it would describe the exact same scalability as $O(n)$.
I appreciate that a lot of these answers are basically telling you to come to this conclusion yourself by reading the definition of Big-O. But this intuitive understanding took me quite a while to wrap my head around and so I lay it out to you as plainly as I can.