Whilst it's not expensive to pay a specialist to submit my website to the search engines, is it not something I could easily do for myself? Would I need potentially expensive, specialist software?
There is no reason why any Webmaster could not handle the submissions for their own website. It's not a complicated task at all. All search engines and directories have a submissions form where you can submit the URL of your website.
For example:
There are of course a great many others, so it can feel like a tedious task. Many people that pay for search engine submissions, do so just to save their own time and sanity. There are also some free search engine submission services available on the web, but these services cannot and do not attempt any kind of submission to the major search engines such as Google, Yahoo or LIVE Search, so you really do get what you pay for in these cases. Nothing of any consequence.
Software is readily available to help you with search engine submissions, some of it is even free. However, these packages still require you to do a lot of the work yourself, but they attempt to help with the process.
If you choose to pay for a search engine submissions to be done for you, the expectation is that the specialist doing the submissions has experience of not only where to submit your website, but also how to ensure your submissions are accepted, e.g. DMOZ, Yahoo Directory, etc. The specialist will also know which search engines not only permit, but promote automated submissions, and of course which one's don't, and where relevant, the frequency for resubmission.
For a business website, it makes sense to have someone experienced handle the submissions for you. However, for a personal / hobby website, why not expand your knowledge and experience by handling it yourself.