You can see it that way. Language is a set of ideas that are replicated by every user that learns it. But the replication isn't perfect and some changes can occur.
New words can be invented, existing words can be altered or become obsolete. Same thing for syntax, accent, sounds, etc. Thus, the average vocabulary and grammar used by a certain generation can be slightly different from the one used by the previous generation. But most of it is the same so you can't really say that two consecutive generations don't speak the same language.
However, after enough generations, the added changes are significant, and the language spoken by generations separated by hundreds (or thousands) of years is clearly not the same anymore. That's how a language becomes a new language.
Are you still with me? Because that's not all. Distance also plays a role.
Imagine a certain change occurs in a language. Let's say people start omitting the sound H. The replication of this phenomenon will mostly occur around the place (or social class) it started, and may or may not spread to the rest of the population.
This way, a language is not spoken exactly the same way everywhere. Within a society, geographical and class accents and dialects can occur.
Now imagine two populations with the same language, but not enough contact to have an homogenous spread of language changes. Different changes will accumulate in the two populations, and the language will become two dialects that are more and more different from each other.
It's even stronger if the two populations are completely isolated from each other. There is absolutely no reason for the exact same changes to occur in the two populations.
After enough time, you get two languages that are different from the original one, but also different from each other.
That's how Latin became Romanian, Italian, French, Spanish, Portuguese and a few other languages that formed until very recently a dialect continuum between Italy and Portugal, for example.
There are other phenomena that can intervene. Change can be accelerated by invasions, changes in society, and even a change of climate (try to have a consonant oriented language in a place with cicadas and enjoy the constant "what did you say?").
Now that I gave a simplified explanation, you are probably thinking "Hey, that's more or less how biological evolution works". And indeed, it's logical. Anything that perpetuates through a certain form of imperfect replication is subjected to a gradual evolution.
This is true for languages, this is true for species and this is true for cultures.