There's actually a library created by Pololu that is quite straightforward to use so you don't need to worry about any of that. I think it covers all available functionality.
I was also a bit confused by the API comment and the datasheet.
I misread the device name so the above is not relevant.
In theory I think to use the API you can just #include the C code they provide into your sketch like a normal library. You can download the API code from the manufacturer ST here.
You'll need to give your name and email but it's free.
It's in C but you can probably understand it if you're familiar with Arduino/C++.
Before compiling you would also need to implement some I2C functions for which they provide prototypes (in the vl53l1_platform.c file). However, I think there are some additional steps when using C files that you will need to take to get this to compile, unfortunately I am not familiar with this so can't help - if anyone does then they can probably write a more complete answer.
The alternative is to write your own functions that replicate the functionality that is described in the API.
I would do this by looking at the basic process flow in the API User Manual, taking each function and working through the API files to see what it does.
The main thing you're looking for is the registers that you need to write (via I2C) in order to trigger a range measurement starting, and then the registers that you need to read to find out the result.
I would note that there are multiple levels of #defines for some of the register values and there is a lot of code that deals with managing the status of the device, so this might take a while.