My understanding of the development of English is that it stems from the Germanic languages, and was then largely influenced by French (and therefore Latin) during the Norman invasion. But I'm not clear on how the Germanic language got their first, especially since I thought parts of the British Isles were under Roman control at one point.
You're correct so far, but missing the chapter between the Roman withdrawal (410AD) and the arrival of the Normans (1066), where, among other things, there was a migration of Germanic peoples from northern Europe into Britain. These groups are typically known as the "Angles, Saxons, and Jutes" (sorry other people!), or the Anglo-Saxons (sorry Jutes!) or even just the Saxons (sorry Angles, although they live on in words like England). Their Germanic language is now referred to as Old English. The subject of their migrations and impact on language in Britain is massive, so I've taken a quick look for some general threads from the last year to get you started.
Some of these posts have been archived, so if you have follow-up questions, you can just ask them here and tag the relevant user's username to notify them
Influence on language
What is the prevailing consensus for the linguistic evolution of the English language after the decline of the Roman Empire? - featuring /u/Flubb
Normans vs Anglo-Saxons: Who Disrupted British Culture/Language/History More? [Tolkien-related] - featuring /u/James123182
Which group prior to the Norman invasion influenced England the most? - featuring /u/alriclofgar
England is called so because of the Angles. So why do we predominately define ourselves pre-Norman conquest as "Saxons" - featuring /u/CorneliusNepos
FAQ section Why doesn't England speak a Latin-based language like other former European Roman provinces
More on the Anglo-Saxons
Where were the Anglo Saxons truly from? - featuring /u/alriclofgar
Why did Britons get defeated by Saxons? - featuring /u/textandtrowel
Who are the Anglo-Saxons in Britain that fought against the Danes? - featuring /u/CheruthCutestory