Snide remarks about the "agency" who left you high and dry aside, who is currently hosting the website? Start there and gain access to the actual website files. Whether you have another agency help you clean this up or you do it yourself, you'll need to have a login and password of some sort. The most common access is FTP.
There's a lot of moving parts here but here's a quick 10,000 foot overview. Like most content management systems (CMS), Craft has 2 components:
- the database (that holds your content)
- the programming "code" (in this case PHP, the software that runs behind the scenes that allows you to make changes and updates using the web based interface)
Craft websites sites are also built using Twig templates, which are, as you presumed, essentially HTML and CSS. You can think of templates as containers for content.
There's probably multiple templates: one for the home page, another for the blog, etc. Depending on what you need to edit, you'll need to find where that content is actually saved. The content that you can edit is stored in the admin section using fields.
Much like Photoshop has multiple layers that store individual pieces of content, fields do much the same way in Craft. Instead of storing a bunch of content in one layer, you break it up so can manipulate it easier.
When you add content in the "admin" area, the template decides where in the template to spit that content out and also how it's presented. If you can't access the content in the admin, it means it's in a template somewhere that you can't edit through the admin screen.
When we build websites for clients, we try to avoid doing that and allow the user to edit the website's content using the admin panel but depending on how they set things up, it sounds like they took some shortcuts.