Replacing the Loop
Use
find -type f -name '*.mkv' -exec ffmpeg -n -i {} -c:v copy ~/Pobrane/{} \;
-exec executes the following ffmpeg command for each of the found paths, replacing {} with the current path. The ; informs find that the ffmpeg command ends there.
Quote '*.mkv' in order to pass the literal string to find, which then searches for files ending with *.mkv. If you do not quote the string and have some mkv files laying around in your working directory, the shell will expand the unquoted *.mkv resulting in find -type f -name firstFile.mkv secondFile.mkv ... before starting find.
Do not quote ~. The unquoted ~ expands to your home directory (probably /home/yourname) but the quoted '~' is a directory/file with the literal name ~ .
Creating Parent Directories
How would I add mkdir -p before the ffmpeg call?
You could wrap the mkdir and ffmpeg in one function and execute the function:
myFunction() {
    mkdir -p "$(dirname "$1")"
    ffmpeg -n -i "$1" -c:v copy ~/Pobrane/"$1"
}
export -f myFunction
find -type f -name '*.mkv' -exec bash -c 'myFunction "$0"' {} \;
or use a loop:
find . -type f -iname "*.txt" -print0 | while IFS= read -r -d $'\0' file; do
    mkdir -p "$(dirname "$file")"
    ffmpeg -n -i "$file" -c:v copy ~/Pobrane/"$file" 
done