split -l 5000 -d --additional-suffix=.txt $FileName file

    -l 5000: split file into files of 5,000 lines each.
    -d: numerical suffix. This will make the suffix go from 00 to 99 by default instead of aa to zz.
    --additional-suffix: lets you specify the suffix, here the extension
    $FileName: name of the file to be split.
    file: prefix to add to the resulting files.

As always, check out man split for more details.
