It is a statement about national accounts that has to hold (an accounting identity). Unless you understand all the terms in the national accounts, the accounting identity may not make sense.
One non-obvious property of the $S=I$ accounting identity is that increases in inventories are considered investment. That is, it is not just fixed investment (machinery, buildings) which people normally associate with “investment.”
Take the simple example of a firm that sells goods with a zero profit margin. Its workers are its customers. It pays its workers \$100, and they save \$10. This gives it sales of \$90, and a build up of inventory of \$10. The savings of the workers did not go to fixed investment, it resulted in a possibly undesired build up of unsold goods at the firm.
One thing to note that we are discussing flows within a single accounting period; in my example, the \$10 build up of inventories happens in the same accounting period as the savings (the same day, for example). As a result, when you write in the question that the \$500 savings today can be used for investment tomorrow is incorrect if we our accounting period is one day at a time - the savings today has to match investment today. (If we had a longer accounting period, such as quarterly, today and tomorrow could end up in the same accounting period.)
As a result, in order to understand the identity, you need to understand the definitions of the various components of the national accounts. If you added an example of a discussion that you had difficulties with to your question, someone might be able to give you more guidance.
My trick to looking at this is to note that the accounting identity holds for any accounting period, including extremely short ones. In particular, any group of transactions satisfies the identity. I then try to see how the low level transactions would end up if they were the only transactions in the national accounts (like the example above).
