Browser dev tools (and most of them really) hook into the application flow to do additional work compared to running it without any developer tools. Depending on what the application does, it can mean that it has to do a lot of extra work and need a lot of extra memory, just to be able to process and store all the extra information that it needs.
I don't know the specifics of why the slowdown is so extreme in this case, usually it has a negligible impact. But I'm guessing it's related to what I wrote above.
Another one that slows a lot of things down is if the application uses the console. Before you open the inspector, those methods are no-ops and just get skipped, basically, but once open, all of those strings have to get copied and it can slow things down quite a bit.
This isn't unique to the web, either, adding the verbose flag to most linux file utilities and then operating on a large set of files will be slower than without the verbose flag, too, just because printing to stdout takes time.
I don't know the specifics of why the slowdown is so extreme in this case, usually it has a negligible impact. But I'm guessing it's related to what I wrote above.