The TLDR for how a bond that continues to pay interest forever can be valued at less than infinity dollars is due to the "time value of money", which states that $X in the future is worth less than $X today. This makes sense intuitively if you consider that if you had that money today, you could invest it and earn interest on it.

So since money in your hands is worth more than that same amount of money in the future, you can actually calculate how much a future cash flow is worth today by discounting it to its present value ("discounted cash flow" aka DCF).

To bring it back to perpetual bonds, if you DCF all of the future cash flows to their present value, you actually get a finite number (due to the diminishing nature of the cash flows that are further and further in the future).

For those who want to learn this in more detail, I recommend MIT's OCW course "Finance Theory I" with Andrew Lo.