If you continue to use a worn chain, then yes, your chances of snapping a chain will increase - So it's important to replace your chain when it is worn - However a chain shouldn't break when it is in a good usable state (ie, not worn). You can tell if your chain is worn by using a
chain wear indicator.
If your chain is worn enough to warrant replacing, you will first need to remove your original chain off your bike by either unclipping the quick-link, or removing the chain with the chain tool if your bike does not have a quick link installed.
Once the original chain is removed, you want to shorten the new chain to the old chain's length with the chain tool, and then fit it to your bike. The easiest way to fit a new chain is with a quick link. If your chain does not come with one, they are very cheap. I bought 2 for £3 the other day.